# Hyperbola · Mathematics · JEE Main

Start Practice## MCQ (Single Correct Answer)

JEE Main 2024 (Online) 9th April Evening Shift

Let the foci of a hyperbola $$H$$ coincide with the foci of the ellipse $$E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$$ and the eccentricity of the hy...

JEE Main 2024 (Online) 8th April Morning Shift

Let $$H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$$ be the hyperbola, whose eccentricity is $$\sqrt{3}$$ and the length of the latus rectum is $$4 \sqrt{3}$...

JEE Main 2024 (Online) 4th April Evening Shift

Consider a hyperbola $$\mathrm{H}$$ having centre at the origin and foci on the $$\mathrm{x}$$-axis. Let $$\mathrm{C}_1$$ be the circle touching the h...

JEE Main 2024 (Online) 1st February Morning Shift

For $0$x^2-y^2 \operatorname{cosec}^2 \theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse $x^2 \operatorname{cosec}^2 \theta+y^2=5$, then the va...

JEE Main 2024 (Online) 31st January Morning Shift

If the foci of a hyperbola are same as that of the ellipse $$\frac{x^2}{9}+\frac{y^2}{25}=1$$ and the eccentricity of the hyperbola is $$\frac{15}{8}$...

JEE Main 2024 (Online) 30th January Evening Shift

Let $$P$$ be a point on the hyperbola $$H: \frac{x^2}{9}-\frac{y^2}{4}=1$$, in the first quadrant such that the area of triangle formed by $$P$$ and t...

JEE Main 2024 (Online) 27th January Evening Shift

Let $$e_1$$ be the eccentricity of the hyperbola $$\frac{x^2}{16}-\frac{y^2}{9}=1$$ and $$e_2$$ be the eccentricity of the ellipse $$\frac{x^2}{a^2}+\...

JEE Main 2023 (Online) 11th April Morning Shift

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...

JEE Main 2023 (Online) 1st February Evening Shift

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 $$\...

JEE Main 2023 (Online) 31st January Evening Shift

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 :

JEE Main 2023 (Online) 25th January Evening Shift

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...

JEE Main 2022 (Online) 28th July Evening Shift

Let the hyperbola $$H: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$$ pass through the point $$(2 \sqrt{2},-2 \sqrt{2})$$. A parabola is drawn whose focu...

JEE Main 2022 (Online) 26th July Evening 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 Morning Shift

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...

JEE Main 2022 (Online) 25th July Evening Shift

Let the foci of the ellipse $$\frac{x^{2}}{16}+\frac{y^{2}}{7}=1$$ and the hyperbola $$\frac{x^{2}}{144}-\frac{y^{2}}{\alpha}=\frac{1}{25}$$ coincide....

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}} \over {{a^2}}} - {{{y^2}} \o...

JEE Main 2022 (Online) 28th June Morning Shift

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 ...

JEE Main 2022 (Online) 26th June Evening Shift

The normal to the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over 9} = 1$$ at the point $$\left( {8,3\sqrt 3 } \right)$$ on it passes through the...

JEE Main 2021 (Online) 26th August Evening Shift

The point $$P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)$$ lies on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ having eccentric...

JEE Main 2021 (Online) 26th August Evening 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) 25th July Morning Shift

The locus of the centroid of the triangle formed by any point P on the hyperbola $$16{x^2} - 9{y^2} + 32x + 36y - 164 = 0$$, and its foci is :

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 parabola y2 = $$\alpha$$x, then $$\al...

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 latus rectum at R(x1, y1), x1 >...

JEE Main 2021 (Online) 16th March Morning Shift

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$...

JEE Main 2021 (Online) 25th February Evening Shift

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 ...

JEE Main 2020 (Online) 5th September Evening 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 circle
x2
+ y2
= 36, then w...

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 at P intersects the x-axis
at ...

JEE Main 2020 (Online) 3rd September Evening Slot

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}} \...

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, then this hyperbola does not
pa...

JEE Main 2020 (Online) 2nd September Evening Slot

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
...

JEE Main 2020 (Online) 2nd September Morning Slot

A line parallel to the straight line 2x – y = 0 is
tangent to the hyperbola
$${{{x^2}} \over 4} - {{{y^2}} \over 2} = 1$$ at the point
$$\left( {{x_1...

JEE Main 2020 (Online) 9th January Morning Slot

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}}...

JEE Main 2020 (Online) 8th January Evening 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 at P is :

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 Morning Slot

If a directrix of a hyperbola centred at the origin and passing through the point (4, –2$$\sqrt 3 $$ ) is 5x = 4$$\sqrt 5 $$ and
its eccentricity is e...

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 a value of m is :

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 the hyperbola at (4,6)
is :

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 following points does not lie on this...

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
eccentricity of the hyperbola is :

JEE Main 2019 (Online) 11th January Evening 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. Then the locus of the centre of...

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) 10th January Morning Slot

The equation of a tangent to the hyperbola 4x2 – 5y2 = 20 parallel to the line x – y = 2 is :

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 x-axis. Then the eccentricity ...

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 }} - {{{y^2}} \over {{{\sin }^2}...

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 $$\sqrt 2 k\,x + k\,y - 4\sqrt 2 = 0$$ (k is any non-zero...

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) then the area (in sq. units)...

JEE Main 2018 (Online) 15th April Evening Slot

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 ...

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 the locus of the mid point of A...

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 $$ \in $$ R), is : ...

JEE Main 2017 (Offline)

A hyperbola passes through the point P$$\left( {\sqrt 2 ,\sqrt 3 } \right)$$ and has foci at $$\left( { \pm 2,0} \right)$$. Then the tangent to this h...

JEE Main 2016 (Online) 10th April Morning Slot

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 ...

JEE Main 2016 (Online) 9th April Morning Slot

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 ...

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 axis is equal to half of the dist...

AIEEE 2007

For the Hyperbola $${{{x^2}} \over {{{\cos }^2}\alpha }} - {{{y^2}} \over {{{\sin }^2}\alpha }} = 1$$ , which of the following remains constant when $...

AIEEE 2007

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...

AIEEE 2005

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...

AIEEE 2003

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 \...

## Numerical

JEE Main 2024 (Online) 8th April Evening Shift

Let $$\mathrm{S}$$ be the focus of the hyperbola $$\frac{x^2}{3}-\frac{y^2}{5}=1$$, on the positive $$x$$-axis. Let $$\mathrm{C}$$ be the circle with ...

JEE Main 2024 (Online) 6th April Evening Shift

The length of the latus rectum and directrices of hyperbola with eccentricity e are 9 and $$x= \pm \frac{4}{\sqrt{3}}$$, respectively. Let the line $$...

JEE Main 2024 (Online) 31st January Morning Shift

Let the foci and length of the latus rectum of an ellipse $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b b e( \pm 5,0)$$ and $$\sqrt{50}$$, respectively. Th...

JEE Main 2024 (Online) 30th January Morning Shift

Let the latus rectum of the hyperbola $$\frac{x^2}{9}-\frac{y^2}{b^2}=1$$ subtend an angle of $$\frac{\pi}{3}$$ at the centre of the hyperbola. If $$\...

JEE Main 2023 (Online) 13th April Evening Shift

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 ...

JEE Main 2023 (Online) 13th April Morning Shift

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}}...

JEE Main 2023 (Online) 11th April Morning Shift

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 ...

JEE Main 2023 (Online) 6th April Evening Shift

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 ...

JEE Main 2023 (Online) 25th January Morning Shift

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...

JEE Main 2022 (Online) 28th July Morning Shift

For the hyperbola $$\mathrm{H}: x^{2}-y^{2}=1$$ and the ellipse $$\mathrm{E}: \frac{x^{2}}{\mathrm{a}^{2}}+\frac{y^{2}}{\mathrm{~b}^{2}}=1$$, a $$>\ma...

JEE Main 2022 (Online) 27th July Evening Shift

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$...

JEE Main 2022 (Online) 27th July Morning Shift

An ellipse $$E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$$ passes through the vertices of the hyperbola $$H: \frac{x^{2}}{49}-\frac{y^{2}}{64}=-1$$....

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 the slope m $$ \in $$ $$(0, \in...

JEE Main 2022 (Online) 29th 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 lengths of the transverse and the c...

JEE Main 2022 (Online) 26th 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 passing through the origin and perpe...

JEE Main 2022 (Online) 25th 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 the equation of the normal at t...

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 length of latus rectum of H is...

JEE Main 2021 (Online) 27th August Evening Shift

Let A (sec$$\theta$$, 2tan$$\theta$$) and B (sec$$\phi$$, 2tan$$\phi$$), where $$\theta$$ + $$\phi$$ = $$\pi$$/2, be two points on the hyperbola 2x2 $...

JEE Main 2021 (Online) 25th February Morning 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 - y - 4\left( {\sqrt 3 } \rig...