Hyperbola PYQs - Last 5 Years

A hyperbola passes through the point $P(\sqrt{2}, \sqrt{3})$ and has foci at $( \pm 2,0)$. Then, the point that lies on the tangent drawn to this hyperbola at $P$ is

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passing through the point $(4,6)$ is 2 , then the equation of the tangent to this hyperbola at $(4,6)$ is

If $\theta$ is the angle subtended by a latus rectum at the centre of the hyperbola having eccentricity $\frac{2}{\sqrt{7}-\sqrt{3}}$, then $\sin \theta=$

The tangent drawn at an extremity (in the first quadrant) of latus rectum of the hyperbola $\frac{x^2}{4}-\frac{y^2}{5}=1$ meets the $X$-axis and $Y$-axis at $A$ and $B$ respectively. If $O$ is the origin, then $(O A)^2-(O B)^2=$

If the angle between the asymptotes of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \tan ^{-1}\left(\frac{2}{3}\right)$ and $a^2-b^2=45$, then $a b=$

Let $P(a \sec \theta, b \tan \theta)$ and $Q(a \sec \phi, b \tan \phi)$, where $\theta+\phi=\frac{\pi}{2}$ be two points on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ If $(h, k)$ is the point of intersection of the normals drawn at $...

If $3 \sqrt{2} x-4 y=12$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{5}{4}$ is its eccentricity, then $a^2-b^2=$

If the normal drawn to the hyperbola $x y=16$ at $(8,2)$ meets the hyperbola again at a point $(\alpha, \beta)$, then $|\beta|+\frac{1}{|\alpha|}=$

If $\theta$ is the acute angle between the tangents drawn from the point $(1,1)$ to the hyperbola $4 x^2-5 y^2-20=0$, then $\tan \theta=$

$x+y+3=0,2 x-y+1=0$ are the equations of the asymptotes of a hyperbola.
If $(1,-2)$ is a point on this hyperbola, then the equation of its conjugate hyperbola is

If $3 x+2 \sqrt{2} y+k=0$ is a normal to the hyperbola $4 x^2-9 y^2-36=0$ making positive intercepts on both the axes, then $k=$

If a hyperbola has asymptotes $3 x-4 y-1=0$ and $4 x-3 y-6=0$, then the transverse and conjugate axes of that hyperbola are

The distance between the tangents of the hyperbola $2 x^2-3 y^2=6$ which are perpendicular to the line $x-2 y+5=0$ is

If a tangent to the hyperbola $x y=-1$ is also a tangent to the parabola $y^2=8 x$, then the equation of that tangent is

By rotating the axes about the origin in anti-clockwise direction with certain angle, if the equation $x^2+4 x y+y^2=1$ is transformed to $\frac{x^2}{a^2}-\frac{y^2}{b^2}=l$, then $\sqrt{\frac{a^2+b^2}{a^2}}=$

If the distance between the foci of a hyperbola $H$ is 26 and distance between its directrices is $\frac{50}{13}$, then the eccentricity of the conjugate hyperbola of the hyperbola $H$ is

If the equation of the tangent of the hyperbola $5 x^2-9 y^2-20 x-18 y-34=0$ which makes an angle $45^{\circ}$ with the positive $X$-axis in positive direction is $x+b y+c=0$, then $b^2+c^2=$

If $\theta$ is the acute angle between the asymptotes of a hyperbola $7 x^2-9 y^2=63$, then $\cos \theta=$

The tangents drawn to the hyperbola $5 x^2-9 y^2=90$ through a variable point $P$ make the angles $\alpha$ and $\beta$ with its transverse axis. If $\alpha, \beta$ are the complementary angles then the locus of $P$ is

If the equation of the hyperbola having $(8,3),(0,3)$ as foci and $\frac{4}{3}$ as eccentricity is $\frac{(x-\alpha)^2}{p}-\frac{(y-\beta)^2}{q}=1$, then $p+q=$

One of the latus recta of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends an angle $2 \tan ^{-1}\left(\frac{3}{2}\right)$ at the centre of the hyperbola. If $b^2=36$ and $e$ is the eccentricity of the given hyperbola, then $\sqrt...

The area of the quadrilateral formed with the foci of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and its conjugate hyperbola is (in sq units)

If $y=x+\sqrt{2}$ is a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{2}=1$, then equations of its directrices are

If $e_1$ and $e_2$ are respectively the eccentricities of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and its conjugate hyperbola, then the line $\frac{x}{2 e_1}+\frac{y}{2 e_2}=1$ touches the circle having centre at the origin, then ...

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\sec \alpha$, then area of the triangle formed by the asymptotes of the hyperbola with any of its tangent is

If the ellipse $4 x^2+9 y^2=36$ is confocal with a hyperbola whose length of the transverse axis is 2 , then the points of intersection of the ellipse and hyperbola lie on the circle

The transformed equation of $x^2-y^2+2 x+4 y=0$ when the origin is shifted to the point $(-1,2)$ is

The line $21 x+5 y=k$ touches the hyperbola $7 x^2-5 y^2=232$, then $k$ is equal to

If the equation $\frac{x^2}{7-k}+\frac{y^2}{5-k}=1$ represents a hyperbola, then

The equation of one of the tangents drawn from the point $(0,1)$ to the hyperbola $45 x^2-4 y^2=5$ is

The equation of the pair of asymptotes of the hyperbola $4 x^2-9 y^2-24 x-36 y-36=0$ is

The descending order of magnitude of the eccentricities of the following hyperbolas is
A. A hyperbola whose distance between foci is three times the distance between its directrices.
B. Hyperbola in which the transverse axis is twice the co...

If a circle of radius 4 cm passes through the foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{4}=1$ and concentric with the hyperbola, then the eccentricity of the conjugate hyperbola of that hyperbola is

If a tangent to the hyperbola $x^2-\frac{y^2}{3}=1$ is also a tangent to the parabola $y^2=8 x$, then equation of such tangent with the positive slope is

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

If $l_1$ and $l_2$ are the lengths of the perpendiculars drawn from a point on the hyperbola $5 x^2-4 y^2-20=0$ to its asymptotes, then $\frac{l_1{ }^2 l_2{ }^2}{100}=$

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

If the product of eccentricities of the ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{16}=-1$ is 1 , then $b^2=$

If the angle between the asymptotes of the hyperbola $x^2-k y^2=3$ is $\frac{\pi}{3}$ and $e$ is its eccentricity, then the pole of the line $x+y-1=0$ with respect to this hyperbola is

If the line $5 x-2 y-6=0$ is a tangent to the hyperbola $5 x^2-k y^2=12$, then the equation of the normal to this hyperbola at the point $(\sqrt{6}, p)(p<0)$ is

If $\theta$ is the acute angle between the tan
from the point $(2,3)$ to the hyperbola from the point $(2,3)$ to the hyperbola $5 x^2-6 y^2-30=0$, then $\tan \theta=$

The difference between the focal distances of any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is 6. If $(\sqrt{13}, k)$ is an end point of a latusrectum of this hyperbola, then $k=$

If $S \equiv \frac{x^2}{k-7}+\frac{y^2}{11-k}-1=0, k \in R-\{7,11\}$, then which one of the following statements is incorrect?

If $A(4,0)$ and $B(-4,0)$ are two points, then the locus of a point $P$ such that $P A-P B=4$ is

Let $(1,2)$ be the focus and $x+y+1=0$ be the directrix of a hyperbola $H$. If $\sqrt{3}$ is the eccentricity of $H$, then its equation is

If \(e_1\) and \(e_2\) are the eccentricities of the hyperbola \(16 x^2-9 y^2=1\) and its conjugate respectively. Then, \(3 e_1=\)

The locus of point of intersection of tangents at the ends of normal chord of the hyperbola \(x^2-y^2=a^2\) is

The value of \(\frac{1+\tan \mathrm{h} x}{1-\tan \mathrm{h} x}\) is

The locus of a variable point whose chord of contact w.r.t. the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\) subtends a right angle at the origin is

Let origin be the centre, \(( \pm 3,0)\) be the foci and \(\frac{3}{2}\) be the eccentricity of a hyperbola.
Then, the line \(2 x-y-1=0\)