Hyperbola — JEE Main PYQs

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 to the hyperbola intersect its conjugate axis at the point …

Let $\mathrm{A}(\sec \theta ,2\tan \theta )$ and $\beta (\sec \varphi ,2\tan \varphi )$, where $\theta +\varphi =\pi /2$ be two points on the hyperbola $2x^2-y^2=2$. If $(\alpha ,\beta )$ is the point of the intersection of the normals to the hyperbola at $A$ and $B$, then …

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

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

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 :

Consider a hyperbola $\mathrm{H}:{\mathrm{x}}^2-2{\mathrm{y}}^2=4$. Let the tangent at a point $\mathrm{P}(4,\sqrt{6})$ meet the x-axis at Q and latus rectum at $\mathrm{R}\left({\mathrm{x}}_1,{\mathrm{y}}_1\right),{\mathrm{x}}_1>0$. If F is a focus of H which is neare …

A square $ABCD$ has all its vertices on the curve $x^2{y}^2=1$. The midpoints of its sides also lie on the same curve. Then, the square of area of $ABCD$ is ____

If a hyperbola passes through the point $P(10,16)$ and it has vertices at $(\pm 6,0)$ then the equation of the normal to it at $P$ is:

If the line $y=mx+c$ 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 of the following is true?

A hyperbola having the transverse axis of length $\sqrt{2}$ has the same foci as that of the ellipse $3x^2+4y^2=12$, then this hyperbola does not pass through which of the following points?

Let $e_1$ and $e_2$ be the eccentricities of the ellipse, $\frac{x^2}{25}+\frac{y^2}{b^2}=1(b<5)$ and the hyperbola, $\frac{x^2}{16}-\frac{y^2}{b^2}=1$ respectively satisfying $e_1{e}_2=1$. If $\alpha$ and $\beta$ are the distances between the foc …

Let $\mathrm{P}(3,3)$ be a point on the hyperbola, $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1$. If the normal to it at P intersects the $x$-axis at $(9,0)$ and $e$ is its eccentricity, then the ordered pair …

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

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

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:

Let $P$ be the point of intersection of the common tangents to the parabola $y^2=12x$ and the hyperbola $8x^2-y^2=8$. If $S$ and $S^{\prime }$ denote the foci of the hyperbola where $S$ lies on the positive $x$-axis then $P$ divides $S{S}^{\prime }$ in a ra …

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

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 of the hyperbola is:

Let $0<\theta <\frac{\pi }{2}$. If the eccentricity of the hyperbola $\frac{x^2}{{\cos }^2\theta }-\frac{y^2}{{\sin }^2\theta }=1$ is greater than 2, then the length of its latus rectum lies in the interval:

If a directrix of a hyperbola centered at the origin and passing through the point $(4,-2\sqrt{3})$ is $5x=4\sqrt{5}$ and its eccentricity is $e$, then

Tangents are drawn to the hyperbola $4x^2-y^2=36$ at the points P and Q . If these tangents intersect at the point $\mathrm{T}(0,3)$ then the area (in sq. units) of $△\mathrm{PTQ}$ is :

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

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal half of the distance between its foci, is:

For the hyperbola $\frac{x^2}{{\cos }^2\alpha }-\frac{y^2}{{\sin }^2\alpha }=1$, which of the following remains constant when $\alpha$ varies?

The locus of a point $P(\alpha ,\beta )$ moving under the condition that the line $y=\alpha x+\beta$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is