JEE Main
Mathematics
Hyperbola
Previous Years Questions

## Numerical

The foci of a hyperbola are $$( \pm 2,0)$$ and its eccentricity is $$\frac{3}{2}$$. A tangent, perpendicular to the line $$2 x+3 y=6$$, is drawn at a ...
Let $$m_{1}$$ and $$m_{2}$$ be the slopes of the tangents drawn from the point $$\mathrm{P}(4,1)$$ to the hyperbola $$H: \frac{y^{2}}{25}-\frac{x^{2}}... Let$$\mathrm{H}_{\mathrm{n}}: \frac{x^{2}}{1+n}-\frac{y^{2}}{3+n}=1, n \in N$$. Let$$\mathrm{k}$$be the smallest even value of$$\mathrm{n}$$such ... Let the eccentricity of an ellipse$$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$is reciprocal to that of the hyperbola$$2 x^{2}-2 y^{2}=1$$. If the ... The vertices of a hyperbola H are ($$\pm$$6, 0) and its eccentricity is$${{\sqrt 5 } \over 2}$$. Let N be the normal to H at a point in the first qu... For the hyperbola$$\mathrm{H}: x^{2}-y^{2}=1$$and the ellipse$$\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$$, a$$>\ma...
A common tangent $$\mathrm{T}$$ to the curves $$\mathrm{C}_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$$ and $$C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1... An ellipse$$E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$passes through the vertices of the hyperbola$$H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$$.... Let the equation of two diameters of a circle$$x^{2}+y^{2}-2 x+2 f y+1=0$$be$$2 p x-y=1$$and$$2 x+p y=4 p$$. Then the slope m$$ \in (0, \in...
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 lengths of the transverse and the c...
Let a line L1 be tangent to the hyperbola $${{{x^2}} \over {16}} - {{{y^2}} \over 4} = 1$$ and let L2 be the line passing through the origin and perpe...
Let the eccentricity of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ be $${5 \over 4}$$. If the equation of the normal at t...
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 length of latus rectum of H is...
Let A (sec$$\theta$$, 2tan$$\theta$$) and B (sec$$\phi$$, 2tan$$\phi$$), where $$\theta$$ + $$\phi$$ = $$\pi$$/2, be two points on the hyperbola 2x2 $... The locus of the point of intersection of the lines $$\left( {\sqrt 3 } \right)kx + ky - 4\sqrt 3 = 0$$ and $$\sqrt 3 x - y - 4\left( {\sqrt 3 } \rig... ## MCQ (Single Correct Answer) Let R be a rectangle given by the lines$$x=0, x=2, y=0$$and$$y=5$$. Let A$$(\alpha,0)$$and B$$(0,\beta),\alpha\in[0,2]$$and$$\beta\in[0,5]$$, be... Let$$\mathrm{P}\left(x_{0}, y_{0}\right)$$be the point on the hyperbola$$3 x^{2}-4 y^{2}=36$$, which is nearest to the line$$3 x+2 y=1$$. Then$$\... Let$\mathrm{H}$be the hyperbola, whose foci are$(1 \pm \sqrt{2}, 0)$and eccentricity is$\sqrt{2}\$. Then the length of its latus rectum is :
Let T and C respectively be the transverse and conjugate axes of the hyperbola $$16{x^2} - {y^2} + 64x + 4y + 44 = 0$$. Then the area of the region ab...
Let the hyperbola $$H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ pass through the point $$(2 \sqrt{2},-2 \sqrt{2})$$. A parabola is drawn whose focu...
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 :
Let the tangent drawn to the parabola $$y^{2}=24 x$$ at the point $$(\alpha, \beta)$$ is perpendicular to the line $$2 x+2 y=5$$. Then the normal to t...
Let the foci of the ellipse $$\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$$ and the hyperbola $$\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}$$ coincide....
Let a > 0, b > 0. Let e and l respectively be the eccentricity and length of the latus rectum of the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \o... Let the eccentricity of the hyperbola$$H:{{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$be$$\sqrt {{5 \over 2}} $$and length of its latus ... The normal to the hyperbola$${{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1$$at the point$$\left( {8,3\sqrt 3 } \right)$$on it passes through the... The point$$P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)$$lies on the hyperbola$${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$having eccentric... The locus of the mid points of the chords of the hyperbola x2$$-$$y2 = 4, which touch the parabola y2 = 8x, is : The locus of the centroid of the triangle formed by any point P on the hyperbola$$16{x^2} - 9{y^2} + 32x + 36y - 164 = 0$$, and its foci is : Let a line L : 2x + y = k, k > 0 be a tangent to the hyperbola x2$$-$$y2 = 3. If L is also a tangent to the parabola y2 =$$\alpha$$x, then$$\al...
Consider a hyperbola H : x2 $$-$$ 2y2 = 4. Let the tangent at a point P(4, $${\sqrt 6 }$$) meet the x-axis at Q and latus rectum at R(x1, y1), x1 >...
The locus of the midpoints of the chord of the circle, x2 + y2 = 25 which is tangent to the hyperbola, $${{{x^2}} \over 9} - {{{y^2}} \over {16}} = 1... A hyperbola passes through the foci of the ellipse$${{{x^2}} \over {25}} + {{{y^2}} \over {16}} = 1$$and its transverse and conjugate axes coincide ... If the line y = mx + c is a common tangent to the hyperbola$${{{x^2}} \over {100}} - {{{y^2}} \over {64}} = 1$$and the circle x2 + y2 = 36, then w... Let P(3, 3) be a point on the hyperbola,$${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$. If the normal to it at P intersects the x-axis at ... Let e1 and e2 be the eccentricities of the ellipse,$${{{x^2}} \over {25}} + {{{y^2}} \over {{b^2}}} = 1$$(b < 5) and the hyperbola,$${{{x^2}} \...
A hyperbola having the transverse axis of length $$\sqrt 2$$ has the same foci as that of the ellipse 3x2 + 4y2 = 12, then this hyperbola does not pa...
For some $$\theta \in \left( {0,{\pi \over 2}} \right)$$, if the eccentricity of the hyperbola, x2–y2sec2$$\theta$$ = 10 is $$\sqrt 5$$ times the ...
A line parallel to the straight line 2x – y = 0 is tangent to the hyperbola $${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1$$ at the point $$\left( {{x_1... If e1 and e2 are the eccentricities of the ellipse,$${{{x^2}} \over {18}} + {{{y^2}} \over 4} = 1$$and the hyperbola,$${{{x^2}} \over 9} - {{{y^2}}...
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 at P is :
If 5x + 9 = 0 is the directrix of the hyperbola 16x2 – 9y2 = 144, then its corresponding focus is :
If a directrix of a hyperbola centred at the origin and passing through the point (4, –2$$\sqrt 3$$ ) is 5x = 4$$\sqrt 5$$ and its eccentricity is e...
If the line y = mx + 7$$\sqrt 3$$ is normal to the hyperbola $${{{x^2}} \over {24}} - {{{y^2}} \over {18}} = 1$$ , then a value of m is :
If the eccentricity of the standard hyperbola passing through the point (4,6) is 2, then the equation of the tangent to the hyperbola at (4,6) is :
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 following points does not lie on this...
If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is :
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. Then the locus of the centre of...
Equation of a common tangent to the parabola y2 = 4x and the hyperbola xy = 2 is :
The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is :
A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length 4 along the x-axis. Then the eccentricity ...
Let $$0 < \theta < {\pi \over 2}$$. If the eccentricity of the hyperbola $${{{x^2}} \over {{{\cos }^2}\theta }} - {{{y^2}} \over {{{\sin }^2}... The locus of the point of intersection of the lines,$$\sqrt 2 x - y + 4\sqrt 2 k = 0$$and$$\sqrt 2 k\,x + k\,y - 4\sqrt 2 = 0$$(k is any non-zero... Tangents are drawn to the hyperbola 4x2 - y2 = 36 at the points P and Q. If these tangents intersect at the point T(0, 3) then the area (in sq. units)... A normal to the hyperbola, 4x2$$-$$9y2 = 36 meets the co-ordinate axes$$x$$and y at A and B, respectively. If the parallelogram OABP (O being the ... If the tangents drawn to the hyperbola 4y2 = x2 + 1 intersect the co-ordinate axes at the distinct points A and B then the locus of the mid point of A... The locus of the point of intersection of the straight lines, tx$$-$$2y$$-$$3t = 0 x$$-$$2ty + 3 = 0 (t$$ \in $$R), is : ... A hyperbola passes through the point P$$\left( {\sqrt 2 ,\sqrt 3 } \right)$$and has foci at$$\left( { \pm 2,0} \right)$$. Then the tangent to this h... A hyperbola whose transverse axis is along the major axis of the conic,$${{{x^2}} \over 3} + {{{y^2}} \over 4} = 4$$and has vertices at the foci of ... Let a and b respectively be the semitransverse and semi-conjugate axes of a hyperbola whose eccentricity satisfies the equation 9e2 − 18e + 5 = 0. If ... The eccentricity of the hyperbola whose length of the latus rectum is equal to$$8$$and the length of its conjugate axis is equal to half of the dist... For the Hyperbola$${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$, which of the following remains constant when ... The normal to a curve at$$P(x,y)$$meets the$$x$$-axis at$$G$$. If the distance of$$G$$from the origin is twice the abscissa of$$P$$, then the c... The locus of a point$$P\left( {\alpha ,\beta } \right)$$moving under the condition that the line$$y = \alpha x + \beta $$is tangent to the hyperbo... The foci of the ellipse$${{{x^2}} \over {16}} + {{{y^2}} \over {{b^2}}} = 1$$and the hyperbola$${{{x^2}} \over {144}} - {{{y^2}} \over {81}} = {1 \...
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