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JEE Main

Conic Sections

Mathematics

Previous Years Questions

MCQ (Single Correct Answer)

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Let the focal chord of the parabola $$\mathrm{P}: y^{2}=4 x$$ along the line $$\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \m...
JEE Main 2022 (Online) 29th July Morning Shift
Let a line L pass through the point of intersection of the lines $$b x+10 y-8=0$$ and $$2 x-3 y=0, \mathrm{~b} \in \math...
JEE Main 2022 (Online) 29th July Morning Shift
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
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
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
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
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
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
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
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
Let the tangent drawn to the parabola $$y^{2}=24 x$$ at the point $$(\alpha, \beta)$$ is perpendicular to the line $$2 x...
JEE Main 2022 (Online) 26th July Morning Shift
Let the foci of the ellipse $$\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
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
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
Let the eccentricity of the ellipse $${x^2} + {a^2}{y^2} = 25{a^2}$$ be b times the eccentricity of the hyperbola $${x^2...
JEE Main 2022 (Online) 30th June Morning Shift
Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C ...
JEE Main 2022 (Online) 24th June Morning Shift
Let the maximum area of the triangle that can be inscribed in the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} ...
JEE Main 2022 (Online) 24th June Evening Shift
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
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
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 $${x^2} + {y...
JEE Main 2022 (Online) 25th June Morning Shift
The line y = x + 1 meets the ellipse $${{{x^2}} \over 4} + {{{y^2}} \over 2} = 1$$ at two points P and Q. If r is the ra...
JEE Main 2022 (Online) 25th June Evening Shift
If the line $$y = 4 + kx,\,k > 0$$, is the tangent to the parabola $$y = x - {x^2}$$ at the point P and V is the vertex ...
JEE Main 2022 (Online) 25th June Evening Shift
The normal to the hyperbola $${{{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
The locus of the mid point of the line segment joining the point (4, 3) and the points on the ellipse $${x^2} + 2{y^2} =...
JEE Main 2022 (Online) 26th June Evening Shift
If m is the slope of a common tangent to the curves $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$ and $${x^2} + {y^2}...
JEE Main 2022 (Online) 26th June Evening Shift
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
Let the eccentricity of an ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$, $$a > b$$, be $${1 \over 4...
JEE Main 2022 (Online) 27th June Morning Shift
If the equation of the parabola, whose vertex is at (5, 4) and the directrix is $$3x + y - 29 = 0$$, is $${x^2} + a{y^2}...
JEE Main 2022 (Online) 27th June Evening Shift
Let the eccentricity of the hyperbola $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $$\sqrt {{5 \over 2...
JEE Main 2022 (Online) 28th June Morning Shift
If vertex of a parabola is (2, $$-$$1) and the equation of its directrix is 4x $$-$$ 3y = 21, then the length of its lat...
JEE Main 2022 (Online) 28th June Evening Shift
Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola $${{{x^2}...
JEE Main 2022 (Online) 28th June Evening Shift
Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of $${\pi \over 4}...
JEE Main 2022 (Online) 29th June Evening Shift
Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of $${\pi \over 2}$$ at the point (3, 0)...
JEE Main 2022 (Online) 29th June Morning Shift
Let $$\theta$$ be the acute angle between the tangents to the ellipse $${{{x^2}} \over 9} + {{{y^2}} \over 1} = 1$$ and ...
JEE Main 2021 (Online) 1st September Evening Shift
Consider the parabola with vertex $$\left( {{1 \over 2},{3 \over 4}} \right)$$ and the directrix $$y = {1 \over 2}$$. Le...
JEE Main 2021 (Online) 1st September Evening Shift
An angle of intersection of the curves, $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ and x2 + y2 = ab, a &g...
JEE Main 2021 (Online) 31st August Evening Shift
The locus of mid-points of the line segments joining ($$-$$3, $$-$$5) and the points on the ellipse $${{{x^2}} \over 4} ...
JEE Main 2021 (Online) 31st August Evening Shift
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
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
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
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
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
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
The point $$P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)$$ lies on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {...
JEE Main 2021 (Online) 26th August Evening Shift
On the ellipse $${{{x^2}} \over 8} + {{{y^2}} \over 4} = 1$$ let P be a point in the second quadrant such that the tange...
JEE Main 2021 (Online) 26th August Morning Shift
Let P and Q be two distinct points on a circle which has center at C(2, 3) and which passes through origin O. If OC is p...
JEE Main 2021 (Online) 27th July Morning Shift
A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this ...
JEE Main 2021 (Online) 27th July Morning Shift
If a tangent to the ellipse x2 + 4y2 = 4 meets the tangents at the extremities of it major axis at B and C, then the cir...
JEE Main 2021 (Online) 25th July Evening Shift
Let an ellipse $$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
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
The locus of the centroid of the triangle formed by any point P on the hyperbola $$16{x^2} - 9{y^2} + 32x + 36y - 164 = ...
JEE Main 2021 (Online) 25th July Morning Shift
Let $${E_1}:{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1,a > b$$. Let E2 be another ellipse such that it tou...
JEE Main 2021 (Online) 22th July Evening Shift
Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2 $$-$$ y2 = 3. If L is also a tangent to the parabol...
JEE Main 2021 (Online) 22th July Evening Shift
Let P be a variable point on the parabola $$y = 4{x^2} + 1$$. Then, the locus of the mid-point of the point P and the fo...
JEE Main 2021 (Online) 20th July Evening Shift
Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola...
JEE Main 2021 (Online) 20th July Morning Shift
Let a tangent be drawn to the ellipse $${{{x^2}} \over {27}} + {y^2} = 1$$ at $$(3\sqrt 3 \cos \theta ,\sin \theta )$$ w...
JEE Main 2021 (Online) 18th March Evening Shift
Consider a hyperbola H : x2 $$-$$ 2y2 = 4. Let the tangent at a point P(4, $${\sqrt 6 }$$) meet the x-axis at Q and latu...
JEE Main 2021 (Online) 18th March Evening Shift
Let L be a tangent line to the parabola y2 = 4x $$-$$ 20 at (6, 2). If L is also a tangent to the ellipse $${{{x^2}} \ov...
JEE Main 2021 (Online) 17th March Evening Shift
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
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
The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, $${{{x^2}} \over ...
JEE Main 2021 (Online) 16th March Morning Shift
If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a $$\ne$$ 0, then 'a' must be greater ...
JEE Main 2021 (Online) 16th March Morning Shift
A hyperbola passes through the foci of the ellipse $${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$ and its transvers...
JEE Main 2021 (Online) 25th February Evening Shift
The shortest distance between the line x $$-$$ y = 1 and the curve x2 = 2y is :
JEE Main 2021 (Online) 25th February Evening Shift
A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points ...
JEE Main 2021 (Online) 25th February Morning Slot
If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x $$-$$ 1, then the co-ordinates o...
JEE Main 2021 (Online) 24th February Evening Slot
The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a moving point of the parab...
JEE Main 2021 (Online) 24th February Morning Slot
The centre of the circle passing through the point (0, 1) and touching the parabola y = x2 at the point (2, 4) is :...
JEE Main 2020 (Online) 6th September Evening Slot
If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentri...
JEE Main 2020 (Online) 6th September Evening Slot
Which of the following points lies on the locus of the foot of perpedicular drawn upon any tangent to the ellipse, $${{{...
JEE Main 2020 (Online) 6th September Morning Slot
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
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
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
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
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, $$...
JEE Main 2020 (Online) 4th September Evening Slot
Let $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ (a > b) be a given ellipse, length of whose latus rectu...
JEE Main 2020 (Online) 4th September Morning Slot
Let P(3, 3) be a point on the hyperbola, $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal to it ...
JEE Main 2020 (Online) 4th September Morning Slot
Let e1 and e2 be the eccentricities of the ellipse, $${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$$(b < 5) a...
JEE Main 2020 (Online) 3rd September Evening Slot
Let the latus ractum of the parabola y2 = 4x be the common chord to the circles C1 and C2 each of them having radius 2...
JEE Main 2020 (Online) 3rd September Evening Slot
Let P be a point on the parabola, y2 = 12x and N be the foot of the perpendicular drawn from P on the axis of the parab...
JEE Main 2020 (Online) 3rd September Morning Slot
A hyperbola having the transverse axis of length $$\sqrt 2 $$ has the same foci as that of the ellipse 3x2 + 4y2 = 12, t...
JEE Main 2020 (Online) 3rd September Morning Slot
For some $$\theta \in \left( {0,{\pi \over 2}} \right)$$, if the eccentricity of the hyperbola, x2–y2sec2$$\theta $$ =...
JEE Main 2020 (Online) 2nd September Evening Slot
The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the ve...
JEE Main 2020 (Online) 2nd September Evening Slot
A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola $${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1...
JEE Main 2020 (Online) 2nd September Morning Slot
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
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
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
If a hyperbola passes through the point P(10, 16) and it has vertices at (± 6, 0), then the equation of the normal to it...
JEE Main 2020 (Online) 8th January Evening Slot
The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy – 3y2 = 0 at the point (2,2) is...
JEE Main 2020 (Online) 8th January Evening Slot
The locus of a point which divides the line segment joining the point (0, –1) and a point on the parabola, x2 = 4y, inte...
JEE Main 2020 (Online) 8th January Morning Slot
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
If 3x + 4y = 12$$\sqrt 2 $$ is a tangent to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 9} = 1$$ for some $$a...
JEE Main 2020 (Online) 7th January Evening Slot
If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12, then the length of i...
JEE Main 2020 (Online) 7th January Morning Slot
If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to:
JEE Main 2020 (Online) 7th January Morning Slot
An ellipse, with foci at (0, 2) and (0, –2) and minor axis of length 4, passes through which of the following points?
JEE Main 2019 (Online) 12th April Evening Slot
The tangents to the curve y = (x – 2)2 – 1 at its points of intersection with the line x – y = 3, intersect at the point...
JEE Main 2019 (Online) 12th April Evening Slot
The equation of common tangent to the curves y2 = 16x and xy = –4, is :
JEE Main 2019 (Online) 12th April Evening Slot
If the normal to the ellipse 3x2 + 4y2 = 12 at a point P on it is parallel to the line, 2x + y = 4 and the tangent to ...
JEE Main 2019 (Online) 12th April Morning Slot
Let P be the point of intersection of the common tangents to the parabola y2 = 12x and the hyperbola 8x2 – y2 = 8. If ...
JEE Main 2019 (Online) 12th April Morning Slot
The tangent and normal to the ellipse 3x2 + 5y2 = 32 at the point P(2, 2) meet the x-axis at Q and R, respectively. Th...
JEE Main 2019 (Online) 10th April Evening Slot
If the line ax + y = c, touches both the curves x2 + y2 = 1 and y2 = 4$$\sqrt 2 $$x , then |c| is equal to : ...
JEE Main 2019 (Online) 10th April Evening Slot
If 5x + 9 = 0 is the directrix of the hyperbola 16x2 – 9y2 = 144, then its corresponding focus is : ...
JEE Main 2019 (Online) 10th April Evening Slot
If the line x – 2y = 12 is tangent to the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$ at the point...
JEE Main 2019 (Online) 10th April Morning Slot
If a directrix of a hyperbola centred at the origin and passing through the point (4, –2$$\sqrt 3 $$ ) is 5x = 4$$\sqrt ...
JEE Main 2019 (Online) 10th April Morning Slot
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
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
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
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
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
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
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
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
The shortest distance between the line y = x and the curve y2 = x – 2 is :
JEE Main 2019 (Online) 8th April Morning Slot
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
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
Let P(4, –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
If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at (–3, 0), then which one of the followi...
JEE Main 2019 (Online) 12th January Morning Slot
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
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
If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x – a2) = 0 and the othertwo verti...
JEE Main 2019 (Online) 11th January Evening Slot
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
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
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
Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is
JEE Main 2019 (Online) 11th January Morning Slot
Let S = $$\left\{ {\left( {x,y} \right) \in {R^2}:{{{y^2}} \over {1 + r}} - {{{x^2}} \over {1 - r}}} \right\};r \ne \pm...
JEE Main 2019 (Online) 10th January Evening Slot
The length of the chord of the parabola x2 $$=$$ 4y having equation x – $$\sqrt 2 y + 4\sqrt 2 = 0$$  is -...
JEE Main 2019 (Online) 10th January Evening Slot
The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is -
JEE Main 2019 (Online) 10th January Morning Slot
If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which on of the following is a valid choice fo...
JEE Main 2019 (Online) 10th January Morning Slot
Let A(4, $$-$$ 4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where ...
JEE Main 2019 (Online) 9th January Evening Slot
A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the ...
JEE Main 2019 (Online) 9th January Evening Slot
If $$\theta $$ denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection,...
JEE Main 2019 (Online) 9th January Morning Slot
Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x is :...
JEE Main 2019 (Online) 9th January Morning Slot
Let $$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
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
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
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
The locus of the point of intersection of the lines, $$\sqrt 2 x - y + 4\sqrt 2 k = 0$$ and $$\sqrt 2 k\,x + k\,y - 4\sq...
JEE Main 2018 (Online) 16th April Morning Slot
A normal to the hyperbola, 4x2 $$-$$ 9y2 = 36 meets the co-ordinate axes $$x$$ and y at A and B, respectively. If the pa...
JEE Main 2018 (Online) 15th April Evening Slot
Tangents drawn from the point ($$-$$8, 0) to the parabola y2 = 8x touch the parabola at $$P$$ and $$Q.$$ If F is the foc...
JEE Main 2018 (Online) 15th April Evening Slot
Two parabolas with a common vertex and with axes along x-axis and $$y$$-axis, respectively intersect each other in the f...
JEE Main 2018 (Online) 15th April Morning Slot
If the tangents drawn to the hyperbola 4y2 = x2 + 1 intersect the co-ordinate axes at the distinct points A and B then t...
JEE Main 2018 (Online) 15th April Morning Slot
Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B,...
JEE Main 2018 (Offline)
Tangents are drawn to the hyperbola 4x2 - y2 = 36 at the points P and Q. If these tangents intersect at the point T(0, 3...
JEE Main 2018 (Offline)
If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then $$\left| c \right|$...
JEE Main 2017 (Online) 9th April Morning Slot
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
Consider an ellipse, whose center is at the origin and its major axis is along the x-axis. If its eccentricity is $${3 \...
JEE Main 2017 (Online) 8th April Morning Slot
If two parallel chords of a circle, having diameter 4units, lie on the opposite sides of the center and subtend angles $...
JEE Main 2017 (Online) 8th April Morning Slot
If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance ...
JEE Main 2017 (Online) 8th April Morning Slot
The locus of the point of intersection of the straight lines, tx $$-$$ 2y $$-$$ 3t = 0 x $$-$$ 2ty + 3 = 0 (t $$ \in $...
JEE Main 2017 (Online) 8th April Morning Slot
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
The eccentricity of an ellipse whose centre is at the origin is $${1 \over 2}$$. If one of its directrices is x = – 4, t...
JEE Main 2017 (Offline)
A hyperbola passes through the point P$$\left( {\sqrt 2 ,\sqrt 3 } \right)$$ and has foci at $$\left( { \pm 2,0} \right)...
JEE Main 2017 (Offline)
A hyperbola whose transverse axis is along the major axis of the conic, $${{{x^2}} \over 3} + {{{y^2}} \over 4} = 4$$ an...
JEE Main 2016 (Online) 10th April Morning Slot
P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the normal at P pass...
JEE Main 2016 (Online) 10th April Morning Slot
Let a and b respectively be the semitransverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the e...
JEE Main 2016 (Online) 9th April Morning Slot
If the tangent at a point on the ellipse $${{{x^2}} \over {27}} + {{{y^2}} \over 3} = 1$$ meets the coordinate axes at A...
JEE Main 2016 (Online) 9th April Morning Slot
Let $$P$$ be the point on the parabola, $${{y^2} = 8x}$$ which is at a minimum distance from the centre $$C$$ of the cir...
JEE Main 2016 (Offline)
The eccentricity of the hyperbola whose length of the latus rectum is equal to $$8$$ and the length of its conjugate axi...
JEE Main 2016 (Offline)
The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse...
JEE Main 2015 (Offline)
Let $$O$$ be the vertex and $$Q$$ be any point on the parabola, $${{x^2} = 8y}$$. If the point $$P$$ divides the line se...
JEE Main 2015 (Offline)
The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0$$, at $$(1,1)$$
JEE Main 2015 (Offline)
The slope of the line touching both the parabolas $${y^2} = 4x$$ and $${x^2} = - 32y$$ is
JEE Main 2014 (Offline)
The locus of the foot of perpendicular drawn from the centre of the ellipse $${x^2} + 3{y^2} = 6$$ on any tangent to it ...
JEE Main 2014 (Offline)
Given : A circle, $$2{x^2} + 2{y^2} = 5$$ and a parabola, $${y^2} = 4\sqrt 5 x$$. Statement-1 : An equation of a common...
JEE Main 2013 (Offline)
The equation of the circle passing through the foci of the ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1$$, and...
JEE Main 2013 (Offline)
An ellipse is drawn by taking a diameter of thec circle $${\left( {x - 1} \right)^2} + {y^2} = 1$$ as its semi-minor axi...
AIEEE 2012
STATEMENT-1 : An equation of a common tangent to the parabola $${y^2} = 16\sqrt 3 x$$ and the ellipse $$2{x^2} + {y^2} =...
AIEEE 2012
Equation of the ellipse whose axes of coordinates and which passes through the point $$(-3,1)$$ and has eccentricity $$\...
AIEEE 2011
If two tangents drawn from a point $$P$$ to the parabola $${y^2} = 4x$$ are at right angles, then the locus of $$P$$ is
AIEEE 2010
The ellipse $${x^2} + 4{y^2} = 4$$ is inscribed in a rectangle aligned with the coordinate axex, which in turn is inscri...
AIEEE 2009
A parabola has the origin as its focus and the line $$x=2$$ as the directrix. Then the vertex of the parabola is at
AIEEE 2008
A focus of an ellipse is at the origin. The directrix is the line $$x=4$$ and the eccentricity is $${{1 \over 2}}$$. The...
AIEEE 2008
The normal to a curve at $$P(x,y)$$ meets the $$x$$-axis at $$G$$. If the distance of $$G$$ from the origin is twice the...
AIEEE 2007
The equation of a tangent to the parabola $${y^2} = 8x$$ is $$y=x+2$$. The point on this line from which the other tange...
AIEEE 2007
For the Hyperbola $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$ , which of the fol...
AIEEE 2007
Angle between the tangents to the curve $$y = {x^2} - 5x + 6$$ at the points $$(2,0)$$ and $$(3,0)$$ is
AIEEE 2006
In the ellipse, the distance between its foci is $$6$$ and minor axis is $$8$$. Then its eccentricity is
AIEEE 2006
The locus of the vertices of the family of parabolas $$y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a$$ is
AIEEE 2006
The locus of a point $$P\left( {\alpha ,\beta } \right)$$ moving under the condition that the line $$y = \alpha x + \bet...
AIEEE 2005
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
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
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
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
The foci of yhe ellipse $${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$ and the hyperbola $${{{x^2}} \over {144}}...
AIEEE 2003
The normal at the point$$\left( {bt_1^2,2b{t_1}} \right)$$ on a parabola meets the parabola again in the point $$\left( ...
AIEEE 2003
Two common tangents to the circle $${x^2} + {y^2} = 2{a^2}$$ and parabola $${y^2} = 8ax$$ are
AIEEE 2002

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
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
For the hyperbola $$\mathrm{H}: x^{2}-y^{2}=1$$ and the ellipse $$\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{...
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A common tangent $$\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
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...
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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 $$H: \frac{x^{...
JEE Main 2022 (Online) 27th July Morning Shift
The sum of diameters of the circles that touch (i) the parabola $$75 x^{2}=64(5 y-3)$$ at the point $$\left(\frac{8}{5},...
JEE Main 2022 (Online) 25th July Morning Shift
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
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 ...
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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
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
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
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
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
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
A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola $$y =...
JEE Main 2022 (Online) 27th June Morning Shift
Let $$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$, a > 0, b > 0, be a hyperbola such that the sum of lengt...
JEE Main 2022 (Online) 29th June Morning Shift
A tangent line L is drawn at the point (2, $$-$$4) on the parabola y2 = 8x. If the line L is also tangent to the circle ...
JEE Main 2021 (Online) 31st August Evening Shift
Let A (sec$$\theta$$, 2tan$$\theta$$) and B (sec$$\phi$$, 2tan$$\phi$$), where $$\theta$$ + $$\phi$$ = $$\pi$$/2, be two...
JEE Main 2021 (Online) 27th August Evening Shift
If the minimum area of the triangle formed by a tangent to the ellipse $${{{x^2}} \over {{b^2}}} + {{{y^2}} \over {4{a^2...
JEE Main 2021 (Online) 27th August Morning Shift
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
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
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
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
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
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
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
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...
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