Q JEE MAIN 2022

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. Then the length of the latus rectum of...

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Q JEE MAIN 2026

The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\frac{\sqrt{3}}{2}$ and its directrices are $...

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Q JEE MAIN 2026

If the eccentricity $e$ of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, passing through $(6,4 \sqrt{3})$, satisfies $15\left(e^2+1\right)=34 e$, then the length of the...

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Q JEE MAIN 2026

Let the eccentricity e of a hyperbola satisfy the equation $6 \mathrm{e}^2-11 \mathrm{e}+3=0$. If the foci of the hyperbola are...

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Q JEE MAIN 2026

Let $e_1$ and $e_2$ be two distinct roots of the equation $x^2-a x+2=0$. Let the sets $\left\{a \in \mathrm{i}: e_1\right.$...

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Q JEE MAIN 2026

Let $\mathrm{H}: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be a hyperbola such that the distance between its foci is 6 and the distance between its...

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Q Jee Main 2026

Let O be the origin, and P and Q be two points on the rectangular hyperbola $x y=12$ such that...

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Q JEE Main 2024

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...

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Q JEE Main 2024

Consider a hyperbola $H$ having centre at the origin and foci and the $x$-axis. Let $C_1$ be the circle touching...

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Q JEE-Main 2024

Let $\mathrm{e}_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=$ 1, $a>b$,...

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Q JEE MAIN 2024

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

JEE Main Physics Easy

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Q JEE MAIN 2026

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $...

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Q JEE MAIN 2026

Let the ellipse $\mathrm{E}: \frac{x^2}{144}+\frac{y^2}{169}=1$ and the hyperbola $\mathrm{H}: \frac{x^2}{16}-\frac{y^2}{\lambda^2}=-1$ have the same foci. If e and L respectively denote...

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Q JEE MAIN 2026

Let PQ be a chord of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$, perpendicular to the $x$-axis such that OPQ is an equilateral triangle,...

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Q JEE MAIN_2026

If the line $\alpha x+2 y=1$, where $\alpha \in \mathbb{R}$, does not meet the hyperbola $x^2-9 y^2=9$, then a possible...

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Q JEE MAIN 2026

Let $\mathrm{P}(10,2 \sqrt{15})$ be a point on the hyperbola $\frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$, whose foci are S and $\mathrm{S}^{\prime}$. If the length of...

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Q JEE MAIN 2026

Let the foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$. If the eccentricity of the hyperbola...

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Q JEE MAIN_2019_

If a directrix of a hyperbola centred at the origin and passing through the point $(4,-2 \sqrt{3})$ is $...

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Q JEE MAIN 2019

If the line $y=m x+7 \sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24}-\frac{y^2}{18}=1$, then a value of $m$ is :

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Q JEE MAIN 2020

For some $\theta \in\left(0, \frac{\pi}{2}\right)$, if the eccentricity of the hyperbola, $x^2-y^2 \sec ^2 \theta=10$ is $\sqrt{5}$ times the eccentricity...

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Q JEE Main 2020

A line parallel to the straight line $2 x-y=0$ is tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{2}=1$ at the point $\left(x_1, y_1\right)$....

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Q JEE MAIN 2019

If the vertices of a hyperbola be at (–2, 0) and (2, 0) and one of its foci be at...

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Q JEE MAIN 2019

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length...

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Q JEE MAIN 2019

A hyperbola has its centre at the origin, passes through the point (4, 2) and has transverse axis of length...

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Q JEE MAIN 2020

If the line $y=m x+c$ is a common tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ and the circle $x^2+y^2=36$, then which one...

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Q JEE MAIN 2019

Equation of a common tangent to the parabola $y^2=4 x$ and the hyperbola $x y=2$ is:

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Q JEE MAIN 2019

If 5 x+9=0 is the directrix of the hyperbola $16 x^2- 9 y^2=144$, then its corresponding focus is :

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Q JEE Main 2019

The equation of a tangent to the hyperbola $4 x^2-5 y^2=20$ parallel to the line $x-y=2$ is

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Q JEE MAIN 2020

Let $\mathrm{P}(3,3)$ be a point on the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal to it at P intersects the x axis...

JEE Main Mathematics Hard

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Q JEE MAIN 2019

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13,...

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Q JEE MAIN 2020

If a hyperbola passes through the point $\mathrm{P}(10,16)$ and it has vertices at $( \pm 6,0)$, then the equation of...

JEE Main Mathematics Medium

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Q JEE-MAIN 2020

Let $e_1$ and $e_2$ be the eccentricities of the ellipse, $\frac{x^2}{25}+\frac{y^2}{b^2}=1(b

JEE Main Mathematics Easy

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Q JEE MAIN_2021_

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

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The locus of the mid points of the chords of the hyperbola $x^2-y^2=4$, which touch the parabola $y^2=8 x$, is...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The point $\mathrm{P}(-2 \sqrt{6}, \sqrt{3})$ lies on the hyperbola $\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ having eccentricity $\frac{\sqrt{5}}{2}$. If the tangent and normal at $P$...

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Q JEE MAIN 2019

If the eccentricity of the standard hyperbola passing through the point (4, 6) is 2, then the equation of the...

JEE Main Mathematics Easy

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Q _JEE MAIN_2021_

The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $...

JEE Main Mathematics Medium

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Q JEE MAIN_2021_

Let a line $L$ : $2 x+y=k, k>0$ be a tangent to the hyperbola $x^2-y^2=3$. If $L$ is also a...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

Consider a hyperbola $H_{\text {in }} x^2 2 y^2=4$. Let the tangent at a point $\mathrm{P}(4, \sqrt{6})$ meet the x...

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Q JEE MAIN 2021

A hyperbola passes through the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and its transverse and conjugate axes coincide with major and...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

Two vertical poles are 150 m apart and the height of one is three times that of the other. If...

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Q JEE MAIN 2021

For which of the following curves, the line $x+\sqrt{3} y=2 \sqrt{3}$ is the tangent at the point $...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The angle of elevation of a jet plane from a point A on the ground is 60°. After a flight...

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Q JEE MAIN 2021

A square $A B C D$ has all its vertices on the curve $x^2 y^2=1$. The midpoints of its sides...

JEE Main Mathematics Medium

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Q JEE MAIN 2021

The locus of the midpoints of the chord of the circle, $x^2+y^2=25$ which is tangent to the hyperbola, $\frac{x^2}{9}-\frac{y^2}{16}=1$ is...

JEE Main Mathematics Easy

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Q JEE MAIN 2021

The locus of the point of intersection of the lines $(\sqrt{3}) k x+k y-4 \sqrt{3}=0$ and $\sqrt{3 x}-y-4(\sqrt{3}) k=0$ is...

JEE Main Mathematics Easy

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Q JEE MAIN_2022

Let the tangent drawn to the parabola $\mathrm{y}^2=24 \mathrm{x}$ at the point $(\alpha, \beta)$ is perpendicular to the line $...

JEE Main Chemistry Easy

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Q JEE MAIN 2022

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 $...

JEE Main Mathematics Medium

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Q JEE MAIN 2022

Let a line $L_1$ be tangent to the hyperbola $\frac{x^2}{16}-\frac{y^2}{4}=1$ and let $L_2$ be the line passing through the origin...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let the hyperbola H : $\frac{x^2}{a^2}-y^2=1$ and the ellipse E : $3 x^2+4 y^2=12$ be such that the length of...

JEE Main Mathematics Easy

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Q JEE MAIN 2022

Let $\mathrm{a}>0, \mathrm{~b}>0$. Let e and $\ell$ respectively be the eccentricity and length of the latus rectum of the hyperbola...

JEE Main Mathematics Hard

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Q JEE MAIN 2022

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be $\frac{5}{4}$. If the equation of the normal at the point $\left(\frac{8}{\sqrt{5}}, \frac{12}{5}\right)$...

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Q JEE MAIN 2022

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

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Q JEE MAIN 2022

Let the eccentricity of the hyperbola $H: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If...

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Q JEE MAIN 2022

Let $\lambda x-2 y=\mu$ be a tangent to the hyperbola $a^2 x^2-y^2=b^2$. Then $\left(\frac{\lambda}{a}\right)^2-\left(\frac{\mu}{b}\right)^2$ is equal to:

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Q JEE MAIN 2023

The foci of a hyperbola are $( \pm 2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line...

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Q JEE MAIN_2025

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be 2 a and 2...

JEE Main Physics Easy

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Q JEE MAIN_2025

Let $c_1$ and $c_2$ be the eccentricities of the ellipse $\frac{x^2}{b^2}+\frac{y^2}{25}=1$ and the hyperbola $\frac{x^2}{16}-\frac{y^2}{b^2}=1$. respectively. If $\mathrm{b}

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Q JEE MAIN 2025

If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3 x^2-y^2-\alpha x+\beta y+\gamma=0$, then $\alpha+\beta+\gamma$ is equal...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the sum of the focal distances of the point $\mathrm{P}(4,3)$ on the hyperbola $\mathrm{H}: \frac{x^2}{\mathrm{a}^2}-\frac{y^2}{\mathrm{~b}^2}=1$ be $8 \sqrt{\frac{5}{3}}$. If...

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4,1)$ to the hyperbola $H: \frac{y^2}{25}-\frac{x^2}{16}=1$....

JEE Main Mathematics Medium

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Q JEE MAIN 2023

Let $H_n=\frac{x^2}{1+n}-\frac{y^2}{3+n}=1, n \in N$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H_k$...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is...

JEE Main Mathematics Medium

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Q JEE MAIN 2024

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

JEE Main Mathematics Medium

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Q JEE MAIN 2024

Let the latus rectum of the hyperbola x^2/9-y^2/b^2 =1 subtend an angle of π/3 at the centre of the hyperbola....

JEE Main Mathematics Medium

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Q JEE-Main 2024

Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latus rectum is $\sqrt{14}$. Then...

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Q JEE-Main 2024

For $0

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Q JEE MAIN 2024

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...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Consider the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ having one of its focus at ${\rm{P}}( - 3,0)$. If the latus...

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Q JEE MAIN 2025

If A and B are the points of intersection of the circle $x^{2}+y^{2}-8 x=0$ and the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$ and a...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the product of the focal distances of the point $P(4,2\sqrt 3)$ on the hyperbola ${\rm{H}}:\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$...

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Q JEE MAIN 2025

Let $\mathrm{H}_{1}: \frac{x^{2}}{\mathrm{a}^{2}}-\frac{y^{2}}{\mathrm{~b}^{2}}=1$ and $\mathrm{H}_{2}:-\frac{x^{2}}{\mathrm{~A}^{2}}+\frac{y^{2}}{\mathrm{~B}^{2}}=1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let...

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Q JEE MAIN 2025

Let one focus of the hyperbola $H:\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ be at $(\sqrt {10} ,0)$ and the corresponding directrix...

JEE Main Mathematics Easy

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Q JEE MAIN 2025

Let the circle C touch the line $x - y + 1 = 0$ , have the centre on the...

JEE Main Mathematics Medium

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Q JEE MAIN 2025

Let the foci of a hyperbola be (1,14) and (1,-12). If it passes through the point(1, 6), then the length...

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Q JEE MAIN 2025

Let $\mathrm{E}: \frac{x^2}{\mathrm{a}^2}+\frac{y^2}{\mathrm{~b}^2}=1, \mathrm{a}>\mathrm{b}$ and $\mathrm{H}: \frac{x^2}{\mathrm{~A}^2}-\frac{y^2}{\mathrm{~B}^2}=1$. Let the distance between the foci of E and the foci of $H$...

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