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Let the focal chord of the parabola $$\mathrm{P}: y^{2}=4 x$$ along the line $$\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \m...
JEE Main 2022 (Online) 29th July Morning Shift
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Let a line L pass through the point of intersection of the lines $$b x+10 y-8=0$$ and $$2 x-3 y=0, \mathrm{~b} \in \math...
JEE Main 2022 (Online) 29th July Morning Shift
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Let the hyperbola $$H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ pass through the point $$(2 \sqrt{2},-2 \sqrt{2})$$. ...
JEE Main 2022 (Online) 28th July Evening Shift
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If the tangents drawn at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the parabola $$y^{2}=2 x-3$$ intersect at the p...
JEE Main 2022 (Online) 28th July Morning Shift
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If the length of the latus rectum of a parabola, whose focus is $$(a, a)$$ and the tangent at its vertex is $$x+y=a$$, i...
JEE Main 2022 (Online) 27th July Evening Shift
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Let $$P(a, b)$$ be a point on the parabola $$y^{2}=8 x$$ such that the tangent at $$P$$ passes through the centre of the...
JEE Main 2022 (Online) 27th July Morning Shift
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If the line $$x-1=0$$ is a directrix of the hyperbola $$k x^{2}-y^{2}=6$$, then the hyperbola passes through the point...
JEE Main 2022 (Online) 26th July Evening Shift
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The equation of a common tangent to the parabolas $$y=x^{2}$$ and $$y=-(x-2)^{2}$$ is
JEE Main 2022 (Online) 26th July Evening Shift
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The acute angle between the pair of tangents drawn to the ellipse $$2 x^{2}+3 y^{2}=5$$ from the point $$(1,3)$$ is...
JEE Main 2022 (Online) 26th July Evening Shift
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Let $$\mathrm{P}$$ and $$\mathrm{Q}$$ be any points on the curves $$(x-1)^{2}+(y+1)^{2}=1$$ and $$y=x^{2}$$, respectivel...
JEE Main 2022 (Online) 26th July Evening Shift
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Let the tangent drawn to the parabola $$y^{2}=24 x$$ at the point $$(\alpha, \beta)$$ is perpendicular to the line $$2 x...
JEE Main 2022 (Online) 26th July Morning Shift
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Let the foci of the ellipse $$\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$$ and the hyperbola $$\frac{x^{2}}{144}-\frac{y^{2}}{\a...
JEE Main 2022 (Online) 25th July Evening Shift
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The tangents at the points $$A(1,3)$$ and $$B(1,-1)$$ on the parabola $$y^{2}-2 x-2 y=1$$ meet at the point $$P$$. Then ...
JEE Main 2022 (Online) 25th July Evening Shift
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If the ellipse $$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ meets the line $$\frac{x}{7}+\frac{y}{2 \sqrt{6}}=1$$ on th...
JEE Main 2022 (Online) 25th July Evening Shift
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Let the eccentricity of the ellipse $${x^2} + {a^2}{y^2} = 25{a^2}$$ be b times the eccentricity of the hyperbola $${x^2...
JEE Main 2022 (Online) 30th June Morning Shift
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Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C ...
JEE Main 2022 (Online) 24th June Morning Shift
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Let the maximum area of the triangle that can be inscribed in the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} ...
JEE Main 2022 (Online) 24th June Evening Shift
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A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the...
JEE Main 2022 (Online) 24th June Evening Shift
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Let $$x = 2t$$, $$y = {{{t^2}} \over 3}$$ be a conic. Let S be the focus and B be the point on the axis of the conic suc...
JEE Main 2022 (Online) 25th June Morning Shift
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If $$y = {m_1}x + {c_1}$$ and $$y = {m_2}x + {c_2}$$, $${m_1} \ne {m_2}$$ are two common tangents of circle $${x^2} + {y...
JEE Main 2022 (Online) 25th June Morning Shift
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The line y = x + 1 meets the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$$ at two points P and Q. If r is the ra...
JEE Main 2022 (Online) 25th June Evening Shift
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If the line $$y = 4 + kx,\,k > 0$$, is the tangent to the parabola $$y = x - {x^2}$$ at the point P and V is the vertex ...
JEE Main 2022 (Online) 25th June Evening Shift
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The normal to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1$$ at the point $$\left( {8,3\sqrt 3 } \rig...
JEE Main 2022 (Online) 26th June Evening Shift
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The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse $${x^2} + 2{y^2} =...
JEE Main 2022 (Online) 26th June Evening Shift
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If m is the slope of a common tangent to the curves $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$ and $${x^2} + {y^2}...
JEE Main 2022 (Online) 26th June Evening Shift
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Let the normal at the point on the parabola y2 = 6x pass through the point (5, $$-$$8). If the tangent at P to the parab...
JEE Main 2022 (Online) 26th June Morning Shift
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Let the eccentricity of an ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $$a > b$$, be $${1 \over 4...
JEE Main 2022 (Online) 27th June Morning Shift
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If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $$3x + y - 29 = 0$$, is $${x^2} + a{y^2}...
JEE Main 2022 (Online) 27th June Evening Shift
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Let the eccentricity of the hyperbola $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $$\sqrt {{5 \over 2...
JEE Main 2022 (Online) 28th June Morning Shift
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If vertex of a parabola is (2, $$-$$1) and the equation of its directrix is 4x $$-$$ 3y = 21, then the length of its lat...
JEE Main 2022 (Online) 28th June Evening Shift
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Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola $${{{x^2}...
JEE Main 2022 (Online) 28th June Evening Shift
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Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of $${\pi \over 4}...
JEE Main 2022 (Online) 29th June Evening Shift
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Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of $${\pi \over 2}$$ at the point (3, 0)...
JEE Main 2022 (Online) 29th June Morning Shift
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Let $$\theta$$ be the acute angle between the tangents to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 1} = 1$$ and ...
JEE Main 2021 (Online) 1st September Evening Shift
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Consider the parabola with vertex $$\left( {{1 \over 2},{3 \over 4}} \right)$$ and the directrix $$y = {1 \over 2}$$. Le...
JEE Main 2021 (Online) 1st September Evening Shift
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An angle of intersection of the curves, $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ and x2 + y2 = ab, a &g...
JEE Main 2021 (Online) 31st August Evening Shift
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The locus of mid-points of the line segments joining ($$-$$3, $$-$$5) and the points on the ellipse $${{{x^2}} \over 4} ...
JEE Main 2021 (Online) 31st August Evening Shift
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The line $$12x\cos \theta + 5y\sin \theta = 60$$ is tangent to which of the following curves?
JEE Main 2021 (Online) 31st August Morning Shift
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The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (&...
JEE Main 2021 (Online) 31st August Morning Shift
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If two tangents drawn from a point P to the parabola y2 = 16(x $$-$$ 3) are at right angles, then the locus of point P i...
JEE Main 2021 (Online) 27th August Evening Shift
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If x2 + 9y2 $$-$$ 4x + 3 = 0, x, y $$\in$$ R, then x and y respectively lie in the intervals :
JEE Main 2021 (Online) 27th August Morning Shift
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A tangent and a normal are drawn at the point P(2, $$-$$4) on the parabola y2 = 8x, which meet the directrix of the para...
JEE Main 2021 (Online) 27th August Morning Shift
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The locus of the mid points of the chords of the hyperbola x2 $$-$$ y2 = 4, which touch the parabola y2 = 8x, is :...
JEE Main 2021 (Online) 26th August Evening Shift
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The point $$P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)$$ lies on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {...
JEE Main 2021 (Online) 26th August Evening Shift
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On the ellipse $${{{x^2}} \over 8} + {{{y^2}} \over 4} = 1$$ let P be a point in the second quadrant such that the tange...
JEE Main 2021 (Online) 26th August Morning Shift
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Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is p...
JEE Main 2021 (Online) 27th July Morning Shift
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A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this ...
JEE Main 2021 (Online) 27th July Morning Shift
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If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of it major axis at B and C, then the cir...
JEE Main 2021 (Online) 25th July Evening Shift
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Let an ellipse $$E:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $${a^2} > {b^2}$$, passes through $$\left...
JEE Main 2021 (Online) 25th July Morning Shift
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Let a parabola b be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origi...
JEE Main 2021 (Online) 25th July Morning Shift
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The locus of the centroid of the triangle formed by any point P on the hyperbola $$16{x^2} - 9{y^2} + 32x + 36y - 164 = ...
JEE Main 2021 (Online) 25th July Morning Shift
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Let $${E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b$$. Let E2 be another ellipse such that it tou...
JEE Main 2021 (Online) 22th July Evening Shift
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Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 $$-$$ y2 = 3. If L is also a tangent to the parabol...
JEE Main 2021 (Online) 22th July Evening Shift
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Let P be a variable point on the parabola $$y = 4{x^2} + 1$$. Then, the locus of the mid-point of the point P and the fo...
JEE Main 2021 (Online) 20th July Evening Shift
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Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola...
JEE Main 2021 (Online) 20th July Morning Shift
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Let a tangent be drawn to the ellipse $${{{x^2}} \over {27}} + {y^2} = 1$$ at $$(3\sqrt 3 \cos \theta ,\sin \theta )$$ w...
JEE Main 2021 (Online) 18th March Evening Shift
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Consider a hyperbola H : x2 $$-$$ 2y2 = 4. Let the tangent at a point P(4, $${\sqrt 6 }$$) meet the x-axis at Q and latu...
JEE Main 2021 (Online) 18th March Evening Shift
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Let L be a tangent line to the parabola y2 = 4x $$-$$ 20 at (6, 2). If L is also a tangent to the ellipse $${{{x^2}} \ov...
JEE Main 2021 (Online) 17th March Evening Shift
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Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equat...
JEE Main 2021 (Online) 16th March Evening Shift
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If the points of intersections of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$ and the circle x2 +...
JEE Main 2021 (Online) 16th March Evening Shift
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The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, $${{{x^2}} \over ...
JEE Main 2021 (Online) 16th March Morning Shift
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If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a $$\ne$$ 0, then 'a' must be greater ...
JEE Main 2021 (Online) 16th March Morning Shift
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A hyperbola passes through the foci of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$ and its transvers...
JEE Main 2021 (Online) 25th February Evening Shift
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The shortest distance between the line x $$-$$ y = 1 and the curve x2 = 2y is :
JEE Main 2021 (Online) 25th February Evening Shift
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A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points ...
JEE Main 2021 (Online) 25th February Morning Slot
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If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x $$-$$ 1, then the co-ordinates o...
JEE Main 2021 (Online) 24th February Evening Slot
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The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a moving point of the parab...
JEE Main 2021 (Online) 24th February Morning Slot
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The centre of the circle passing through the point (0, 1) and touching the parabola y = x2 at the point (2, 4) is :...
JEE Main 2020 (Online) 6th September Evening Slot
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If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentri...
JEE Main 2020 (Online) 6th September Evening Slot
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Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent to the ellipse, $${{{...
JEE Main 2020 (Online) 6th September Morning Slot
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Let L1 be a tangent to the parabola y2 = 4(x + 1) and L2 be a tangent to the parabola y2 = 8(x + 2) such that L1 and ...
JEE Main 2020 (Online) 6th September Morning Slot
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If the line y = mx + c is a common tangent to the hyperbola $${{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1$$ and the...
JEE Main 2020 (Online) 5th September Evening Slot
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If the co-ordinates of two points A and B are $$\left( {\sqrt 7 ,0} \right)$$ and $$\left( { - \sqrt 7 ,0} \right)$$ res...
JEE Main 2020 (Online) 5th September Morning Slot
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If the common tangent to the parabolas, y2 = 4x and x2 = 4y also touches the circle, x2 + y2 = c2, then c is equal to :...
JEE Main 2020 (Online) 5th September Morning Slot
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Let x = 4 be a directrix to an ellipse whose centre is at the origin and its eccentricity is $${1 \over 2}$$. If P(1, $$...
JEE Main 2020 (Online) 4th September Evening Slot
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Let $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ (a > b) be a given ellipse, length of whose latus rectu...
JEE Main 2020 (Online) 4th September Morning Slot
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Let P(3, 3) be a point on the hyperbola, $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal to it ...
JEE Main 2020 (Online) 4th September Morning Slot
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Let e1 and e2 be the eccentricities of the ellipse, $${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$$(b < 5) a...
JEE Main 2020 (Online) 3rd September Evening Slot
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Let the latus ractum of the parabola y2 = 4x be the common chord to the circles C1 and C2 each of them having radius 2...
JEE Main 2020 (Online) 3rd September Evening Slot
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Let P be a point on the parabola, y2 = 12x and N be the foot of the perpendicular drawn from P on the axis of the parab...
JEE Main 2020 (Online) 3rd September Morning Slot
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A hyperbola having the transverse axis of length $$\sqrt 2 $$ has the same foci as that of the ellipse 3x2 + 4y2 = 12, t...
JEE Main 2020 (Online) 3rd September Morning Slot
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For some $$\theta \in \left( {0,{\pi \over 2}} \right)$$, if the eccentricity of the hyperbola, x2â€“y2sec2$$\theta $$ =...
JEE Main 2020 (Online) 2nd September Evening Slot
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The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the ve...
JEE Main 2020 (Online) 2nd September Evening Slot
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A line parallel to the straight line 2x â€“ y = 0 is tangent to the hyperbola $${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1...
JEE Main 2020 (Online) 2nd September Morning Slot
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If one end of a focal chord AB of the parabola y2 = 8x is at $$A\left( {{1 \over 2}, - 2} \right)$$, then the equation o...
JEE Main 2020 (Online) 9th January Evening Slot
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The length of the minor axis (along y-axis) of an ellipse in the standard form is $${4 \over {\sqrt 3 }}$$. If this elli...
JEE Main 2020 (Online) 9th January Evening Slot
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If e1 and e2 are the eccentricities of the ellipse, $${{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1$$ and the hyperbola, ...
JEE Main 2020 (Online) 9th January Morning Slot
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If a hyperbola passes through the point P(10, 16) and it has vertices at (Â± 6, 0), then the equation of the normal to it...
JEE Main 2020 (Online) 8th January Evening Slot
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The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy â€“ 3y2 = 0 at the point (2,2) is...
JEE Main 2020 (Online) 8th January Evening Slot
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The locus of a point which divides the line segment joining the point (0, â€“1) and a point on the parabola, x2 = 4y, inte...
JEE Main 2020 (Online) 8th January Morning Slot
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Let the line y = mx and the ellipse 2x2 + y2 = 1 intersect at a ponit P in the first quadrant. If the normal to this ell...
JEE Main 2020 (Online) 8th January Morning Slot
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If 3x + 4y = 12$$\sqrt 2 $$ is a tangent to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1$$ for some $$a...
JEE Main 2020 (Online) 7th January Evening Slot
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If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of i...
JEE Main 2020 (Online) 7th January Morning Slot
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If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to:
JEE Main 2020 (Online) 7th January Morning Slot
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An ellipse, with foci at (0, 2) and (0, â€“2) and minor axis of length 4, passes through which of the following points?
JEE Main 2019 (Online) 12th April Evening Slot
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The tangents to the curve y = (x â€“ 2)2 â€“ 1 at its points of intersection with the line x â€“ y = 3, intersect at the point...
JEE Main 2019 (Online) 12th April Evening Slot
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The equation of common tangent to the curves y2 = 16x and xy = â€“4, is :
JEE Main 2019 (Online) 12th April Evening Slot
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If the normal to the ellipse 3x2 + 4y2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to ...
JEE Main 2019 (Online) 12th April Morning Slot
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Let P be the point of intersection of the common tangents to the parabola y2 = 12x and the hyperbola 8x2 â€“ y2 = 8. If ...
JEE Main 2019 (Online) 12th April Morning Slot
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The tangent and normal to the ellipse 3x2 + 5y2 = 32 at the point P(2, 2) meet the x-axis at Q and R, respectively. Th...
JEE Main 2019 (Online) 10th April Evening Slot
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If the line ax + y = c, touches both the curves x2 + y2 = 1 and y2 = 4$$\sqrt 2 $$x , then |c| is equal to : ...
JEE Main 2019 (Online) 10th April Evening Slot
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If 5x + 9 = 0 is the directrix of the hyperbola 16x2 â€“ 9y2 = 144, then its corresponding focus is : ...
JEE Main 2019 (Online) 10th April Evening Slot
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If the line x â€“ 2y = 12 is tangent to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ at the point...
JEE Main 2019 (Online) 10th April Morning Slot
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If a directrix of a hyperbola centred at the origin and passing through the point (4, â€“2$$\sqrt 3 $$ ) is 5x = 4$$\sqrt ...
JEE Main 2019 (Online) 10th April Morning Slot
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The area (in sq. units) of the smaller of the two circles that touch the parabola, y2 = 4x at the point (1, 2) and the ...
JEE Main 2019 (Online) 9th April Evening Slot
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If the tangent to the parabola y2 = x at a point ($$\alpha $$, $$\beta $$), ($$\beta $$ > 0) is also a tangent to the...
JEE Main 2019 (Online) 9th April Evening Slot
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If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is
JEE Main 2019 (Online) 9th April Morning Slot
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If the line y = mx + 7$$\sqrt 3 $$ is normal to the hyperbola $${{{x^2}} \over {24}} - {{{y^2}} \over {18}} = 1$$ , then...
JEE Main 2019 (Online) 9th April Morning Slot
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In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of...
JEE Main 2019 (Online) 8th April Evening Slot
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The tangent to the parabola y2 = 4x at the point where it intersects the circle x2 + y2 = 5 in the first quadrant, pa...
JEE Main 2019 (Online) 8th April Evening Slot
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If the eccentricity of the standard hyperbola passing through the point (4,6) is 2, then the equation of the tangent to ...
JEE Main 2019 (Online) 8th April Evening Slot
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If the tangents on the ellipse 4x2 + y2 = 8 at the points (1, 2) and (a, b) are perpendicular to each other, then a2 is ...
JEE Main 2019 (Online) 8th April Morning Slot
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The shortest distance between the line y = x and the curve y2 = x â€“ 2 is :
JEE Main 2019 (Online) 8th April Morning Slot
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Let S and S' be the foci of an ellipse and B be any one of the extremities of its minor axis. If $$\Delta $$S'BS is a ri...
JEE Main 2019 (Online) 12th January Evening Slot
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The equation of a tangent to the parabola, x2 = 8y, which makes an angle $$\theta $$ with the positive directions of x-...
JEE Main 2019 (Online) 12th January Evening Slot
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Let P(4, â€“4) and Q(9, 6) be two points on the parabola, y2 = 4x and let x be any point on the arc POQ of this parabola, ...
JEE Main 2019 (Online) 12th January Morning Slot
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If the vertices of a hyperbola be at (â€“2, 0) and (2, 0) and one of its foci be at (â€“3, 0), then which one of the followi...
JEE Main 2019 (Online) 12th January Morning Slot
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The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola,...
JEE Main 2019 (Online) 12th January Morning Slot
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A circle cuts a chord of length 4a on the x-axis and passes through a point on the y-axis, distant 2b from the origin. T...
JEE Main 2019 (Online) 11th January Evening Slot
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If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x â€“ a2) = 0 and the othertwo verti...
JEE Main 2019 (Online) 11th January Evening Slot
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If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricit...
JEE Main 2019 (Online) 11th January Evening Slot
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Let the length of the latus rectum of an ellipse with its major axis along x-axis and centre at the origin, be 8. If the...
JEE Main 2019 (Online) 11th January Evening Slot
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If tangents are drawn to the ellipse x2 + 2y2 = 2 at all points on the ellipse other than its four vertices then the m...
JEE Main 2019 (Online) 11th January Morning Slot
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Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is
JEE Main 2019 (Online) 11th January Morning Slot
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Let S = $$\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm...
JEE Main 2019 (Online) 10th January Evening Slot
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The length of the chord of the parabola x2 $$=$$ 4y having equation x â€“ $$\sqrt 2 y + 4\sqrt 2 = 0$$ is -...
JEE Main 2019 (Online) 10th January Evening Slot
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The equation of a tangent to the hyperbola 4x2 â€“ 5y2 = 20 parallel to the line x â€“ y = 2 is -
JEE Main 2019 (Online) 10th January Morning Slot
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If the parabolas y2 = 4b(x â€“ c) and y2 = 8ax have a common normal, then which on of the following is a valid choice fo...
JEE Main 2019 (Online) 10th January Morning Slot
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Let A(4, $$-$$ 4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where ...
JEE Main 2019 (Online) 9th January Evening Slot
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A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the ...
JEE Main 2019 (Online) 9th January Evening Slot
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If $$\theta $$ denotes the acute angle between the curves, y = 10 â€“ x2 and y = 2 + x2 at a point of their intersection,...
JEE Main 2019 (Online) 9th January Morning Slot
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Equation of a common tangent to the circle, x2 + y2 â€“ 6x = 0 and the parabola, y2 = 4x is :...
JEE Main 2019 (Online) 9th January Morning Slot
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Let $$0 < \theta < {\pi \over 2}$$. If the eccentricity of the hyperbola $${{{x^2}} \over {{{\cos }^2}\theta }}...
JEE Main 2019 (Online) 9th January Morning Slot
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Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on ...
JEE Main 2019 (Online) 9th January Morning Slot
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Let P be a point on the parabola, x2 = 4y. If the distance of P from the center of the circle, x2 + y2 + 6x + 8 = 0 is m...
JEE Main 2018 (Online) 16th April Morning Slot
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If the length of the latus rectum of an ellipse is 4 units and the distance between a focus an its nearest vertex on the...
JEE Main 2018 (Online) 16th April Morning Slot
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The locus of the point of intersection of the lines, $$\sqrt 2 x - y + 4\sqrt 2 k = 0$$ and $$\sqrt 2 k\,x + k\,y - 4\sq...
JEE Main 2018 (Online) 16th April Morning Slot
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A normal to the hyperbola, 4x2 $$-$$ 9y2 = 36 meets the co-ordinate axes $$x$$ and y at A and B, respectively. If the pa...
JEE Main 2018 (Online) 15th April Evening Slot
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Tangents drawn from the point ($$-$$8, 0) to the parabola y2 = 8x touch the parabola at $$P$$ and $$Q.$$ If F is the foc...
JEE Main 2018 (Online) 15th April Evening Slot
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Two parabolas with a common vertex and with axes along x-axis and $$y$$-axis, respectively intersect each other in the f...
JEE Main 2018 (Online) 15th April Morning Slot
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If the tangents drawn to the hyperbola 4y2 = x2 + 1 intersect the co-ordinate axes at the distinct points A and B then t...
JEE Main 2018 (Online) 15th April Morning Slot
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Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B,...
JEE Main 2018 (Offline)
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Tangents are drawn to the hyperbola 4x2 - y2 = 36 at the points P and Q. If these tangents intersect at the point T(0, 3...
JEE Main 2018 (Offline)
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If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then $$\left| c \right|$...
JEE Main 2017 (Online) 9th April Morning Slot
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The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the poin...
JEE Main 2017 (Online) 9th April Morning Slot
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Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is $${3 \...
JEE Main 2017 (Online) 8th April Morning Slot
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If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles $...
JEE Main 2017 (Online) 8th April Morning Slot
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If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance ...
JEE Main 2017 (Online) 8th April Morning Slot
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The locus of the point of intersection of the straight lines, tx $$-$$ 2y $$-$$ 3t = 0 x $$-$$ 2ty + 3 = 0 (t $$ \in $...
JEE Main 2017 (Online) 8th April Morning Slot
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Let z$$ \in $$C, the set of complex numbers. Then the equation, 2|z + 3i| $$-$$ |z $$-$$ i| = 0 represents :
JEE Main 2017 (Online) 8th April Morning Slot
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The eccentricity of an ellipse whose centre is at the origin is $${1 \over 2}$$. If one of its directrices is x = â€“ 4, t...
JEE Main 2017 (Offline)
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A hyperbola passes through the point P$$\left( {\sqrt 2 ,\sqrt 3 } \right)$$ and has foci at $$\left( { \pm 2,0} \right)...
JEE Main 2017 (Offline)
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A hyperbola whose transverse axis is along the major axis of the conic, $${{{x^2}} \over 3} + {{{y^2}} \over 4} = 4$$ an...
JEE Main 2016 (Online) 10th April Morning Slot
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P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the normal at P pass...
JEE Main 2016 (Online) 10th April Morning Slot
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Let a and b respectively be the semitransverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the e...
JEE Main 2016 (Online) 9th April Morning Slot
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If the tangent at a point on the ellipse $${{{x^2}} \over {27}} + {{{y^2}} \over 3} = 1$$ meets the coordinate axes at A...
JEE Main 2016 (Online) 9th April Morning Slot
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Let $$P$$ be the point on the parabola, $${{y^2} = 8x}$$ which is at a minimum distance from the centre $$C$$ of the cir...
JEE Main 2016 (Offline)
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The eccentricity of the hyperbola whose length of the latus rectum is equal to $$8$$ and the length of its conjugate axi...
JEE Main 2016 (Offline)
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The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse...
JEE Main 2015 (Offline)
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Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $${{x^2} = 8y}$$. If the point $$P$$ divides the line se...
JEE Main 2015 (Offline)
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The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0$$, at $$(1,1)$$
JEE Main 2015 (Offline)
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The slope of the line touching both the parabolas $${y^2} = 4x$$ and $${x^2} = - 32y$$ is
JEE Main 2014 (Offline)
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The locus of the foot of perpendicular drawn from the centre of the ellipse $${x^2} + 3{y^2} = 6$$ on any tangent to it ...
JEE Main 2014 (Offline)
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Given : A circle, $$2{x^2} + 2{y^2} = 5$$ and a parabola, $${y^2} = 4\sqrt 5 x$$. Statement-1 : An equation of a common...
JEE Main 2013 (Offline)
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The equation of the circle passing through the foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and...
JEE Main 2013 (Offline)
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An ellipse is drawn by taking a diameter of thec circle $${\left( {x - 1} \right)^2} + {y^2} = 1$$ as its semi-minor axi...
AIEEE 2012
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STATEMENT-1 : An equation of a common tangent to the parabola $${y^2} = 16\sqrt 3 x$$ and the ellipse $$2{x^2} + {y^2} =...
AIEEE 2012
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Equation of the ellipse whose axes of coordinates and which passes through the point $$(-3,1)$$ and has eccentricity $$\...
AIEEE 2011
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If two tangents drawn from a point $$P$$ to the parabola $${y^2} = 4x$$ are at right angles, then the locus of $$P$$ is
AIEEE 2010
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The ellipse $${x^2} + 4{y^2} = 4$$ is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscri...
AIEEE 2009
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A parabola has the origin as its focus and the line $$x=2$$ as the directrix. Then the vertex of the parabola is at
AIEEE 2008
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A focus of an ellipse is at the origin. The directrix is the line $$x=4$$ and the eccentricity is $${{1 \over 2}}$$. The...
AIEEE 2008
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The normal to a curve at $$P(x,y)$$ meets the $$x$$-axis at $$G$$. If the distance of $$G$$ from the origin is twice the...
AIEEE 2007
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The equation of a tangent to the parabola $${y^2} = 8x$$ is $$y=x+2$$. The point on this line from which the other tange...
AIEEE 2007
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For the Hyperbola $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$ , which of the fol...
AIEEE 2007
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Angle between the tangents to the curve $$y = {x^2} - 5x + 6$$ at the points $$(2,0)$$ and $$(3,0)$$ is
AIEEE 2006
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In the ellipse, the distance between its foci is $$6$$ and minor axis is $$8$$. Then its eccentricity is
AIEEE 2006
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The locus of the vertices of the family of parabolas $$y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a$$ is
AIEEE 2006
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The locus of a point $$P\left( {\alpha ,\beta } \right)$$ moving under the condition that the line $$y = \alpha x + \bet...
AIEEE 2005
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Let $$P$$ be the point $$(1, 0)$$ and $$Q$$ a point on the parabola $${y^2} = 8x$$. The locus of mid point of $$PQ$$ is
AIEEE 2005
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An ellipse has $$OB$$ as semi minor axis, $$F$$ and $$F$$' its focii and theangle $$FBF$$' is a right angle. Then the ec...
AIEEE 2005
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The eccentricity of an ellipse, with its centre at the origin, is $${1 \over 2}$$. If one of the directrices is $$x=4$$,...
AIEEE 2004
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If $$a \ne 0$$ and the line $$2bx+3cy+4d=0$$ passes through the points of intersection of the parabolas $${y^2} = 4ax$$ ...
AIEEE 2004
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The foci of yhe ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$ and the hyperbola $${{{x^2}} \over {144}}...
AIEEE 2003
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The normal at the point$$\left( {bt_1^2,2b{t_1}} \right)$$ on a parabola meets the parabola again in the point $$\left( ...
AIEEE 2003
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Two common tangents to the circle $${x^2} + {y^2} = 2{a^2}$$ and parabola $${y^2} = 8ax$$ are
AIEEE 2002
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Numerical
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Two tangent lines $$l_{1}$$ and $$l_{2}$$ are drawn from the point $$(2,0)$$ to the parabola $$2 \mathrm{y}^{2}=-x$$. If...
JEE Main 2022 (Online) 28th July Evening Shift
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Let the tangents at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the ellipse $$\frac{x^{2}}{2}+\frac{y^{2}}{4}=1$$ me...
JEE Main 2022 (Online) 28th July Evening Shift
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For the hyperbola $$\mathrm{H}: x^{2}-y^{2}=1$$ and the ellipse $$\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{...
JEE Main 2022 (Online) 28th July Morning Shift
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A common tangent $$\mathrm{T}$$ to the curves $$\mathrm{C}_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$ and $$C_{2}: \frac{x...
JEE Main 2022 (Online) 27th July Evening Shift
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If the length of the latus rectum of the ellipse $$x^{2}+4 y^{2}+2 x+8 y-\lambda=0$$ is 4 , and $$l$$ is the length of i...
JEE Main 2022 (Online) 27th July Morning Shift
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An ellipse $$E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ passes through the vertices of the hyperbola $$H: \frac{x^{...
JEE Main 2022 (Online) 27th July Morning Shift
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The sum of diameters of the circles that touch (i) the parabola $$75 x^{2}=64(5 y-3)$$ at the point $$\left(\frac{8}{5},...
JEE Main 2022 (Online) 25th July Morning Shift
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Let the equation of two diameters of a circle $$x^{2}+y^{2}-2 x+2 f y+1=0$$ be $$2 p x-y=1$$ and $$2 x+p y=4 p$$. Then t...
JEE Main 2022 (Online) 25th July Morning Shift
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Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times ...
JEE Main 2022 (Online) 30th June Morning Shift
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If two tangents drawn from a point ($$\alpha$$, $$\beta$$) lying on the ellipse 25x2 + 4y2 = 1 to the parabola y2 = 4x a...
JEE Main 2022 (Online) 24th June Morning Shift
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Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6....
JEE Main 2022 (Online) 24th June Evening Shift
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Let the hyperbola $$H:{{{x^2}} \over {{a^2}}} - {y^2} = 1$$ and the ellipse $$E:3{x^2} + 4{y^2} = 12$$ be such that the ...
JEE Main 2022 (Online) 24th June Evening Shift
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Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $${5 \over 4}$$. If t...
JEE Main 2022 (Online) 25th June Evening Shift
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Let a line L1 be tangent to the hyperbola $${{{x^2}} \over {16}} - {{{y^2}} \over 4} = 1$$ and let L2 be the line passin...
JEE Main 2022 (Online) 26th June Evening Shift
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Let the common tangents to the curves $$4({x^2} + {y^2}) = 9$$ and $${y^2} = 4x$$ intersect at the point Q. Let an ellip...
JEE Main 2022 (Online) 26th June Morning Shift
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A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola $$y =...
JEE Main 2022 (Online) 27th June Morning Shift
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Let $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, a > 0, b > 0, be a hyperbola such that the sum of lengt...
JEE Main 2022 (Online) 29th June Morning Shift
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A tangent line L is drawn at the point (2, $$-$$4) on the parabola y2 = 8x. If the line L is also tangent to the circle ...
JEE Main 2021 (Online) 31st August Evening Shift
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Let A (sec$$\theta$$, 2tan$$\theta$$) and B (sec$$\phi$$, 2tan$$\phi$$), where $$\theta$$ + $$\phi$$ = $$\pi$$/2, be two...
JEE Main 2021 (Online) 27th August Evening Shift
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If the minimum area of the triangle formed by a tangent to the ellipse $${{{x^2}} \over {{b^2}}} + {{{y^2}} \over {4{a^2...
JEE Main 2021 (Online) 27th August Morning Shift
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Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, $$-$$4), one focus at (4,...
JEE Main 2021 (Online) 27th July Evening Shift
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If the point on the curve y2 = 6x, nearest to the point $$\left( {3,{3 \over 2}} \right)$$ is ($$\alpha$$, $$\beta$$), t...
JEE Main 2021 (Online) 20th July Evening Shift
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Let y = mx + c, m > 0 be the focal chord of y2 = $$-$$ 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value o...
JEE Main 2021 (Online) 20th July Morning Shift
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Let L be a common tangent line to the curves 4x2 + 9y2 = 36 and (2x)2 + (2y)2 = 31. Then the square of the slope of the ...
JEE Main 2021 (Online) 26th February Evening Shift
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A line is a common tangent to the circle (x $$-$$ 3)2 + y2 = 9 and the parabola y2 = 4x. If the two points of contact (a...
JEE Main 2021 (Online) 25th February Evening Shift
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The locus of the point of intersection of the lines $$\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0$$ and $$\sqrt 3 x...
JEE Main 2021 (Online) 25th February Morning Slot
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If the tangent to the curve, y = ex at a point (c, ec) and the normal to the parabola, y2 = 4x at the point (1, 2) inte...
JEE Main 2020 (Online) 3rd September Evening Slot
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Let a line y = mx (m > 0) intersect the parabola, y2 = x at a point P, other than the origin. Let the tangent to it a...
JEE Main 2020 (Online) 8th January Evening Slot
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