Hyperbola practice · Mixed difficulty

Tangents are drawn to the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ parallel to the straight line $2 x-y=1$. The points of contact of the tangents on the hyperbola, are

If the tangent at $\left( {\alpha ,\beta } \right)$ to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ cuts the auxiliary circle at points whose ordinates are ${y_1}$ and ${y_2}$, then $\frac{1}{{{y_1}}} + \frac{1}{{{y_2}}} =$

The equation of the director circle of the hyperbola $9{x^2} - 16{y^2} = 144$ is

The foci of a hyperbola are $(-5,18)$ and $(10,20)$ and it touches the $y$-axis. The length of its transverse axis, is

The equation of the hyperbola whose foci are $\left( {6,5} \right),\left( { - 4,5} \right)$ and eccentricity $\frac{5}{4}$, is

For the ellipse $3{x^2} + 4{y^2} + 6x - 8y - 5 = 0$ the eccentricity is

Given the points $A\left( {0,4} \right)$ and $B\left( {0, - 4} \right)$, then the equation of the locus of the point $P\left( {x,y} \right)$ such that, $\left| {AP - BP} \right| = 6$, is

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

The line $9 \sqrt{3} x+12 y=234 \sqrt{3}$ is a normal to the hyperbola $\frac{x^2}{81}-\frac{y^2}{36}=1$ at the points.

The equation of the normal to the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ at the point $(8,3 \sqrt{3})$ is

The product of the lengths of perpendiculars drawn from any point on the hyperbola ${x^2} - 2{y^2} - 2 = 0$ to its asymptotes is

The normal at $P$ to a hyperbola of eccentricity $e$, intersects its transverse and conjugate axes at $L$ and $M$ respectively. If locus of the mid point of $LM$ is hyperbola, then eccentricity of the hyperbola is

If $lx + my + n = 0$ is a tangent to the rectangular hyperbola $xy = {c^2},$ then

Focus of hyperbola is $\left( { \pm 3,0} \right)$ and equation of tangent is $2x + y - 4 = 0$, find the equation of hyperbola

If length of the transverse axis of a hyperbola is 8 and its eccentricity is $\sqrt{5} / 2$ then the length of a latus rectum of the hyperbola is

The eccentricity of the hyperbola in the standard form $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, passing through (3,0) and $\left( {3,\sqrt 2 ,2} \right)$ is

The eccentricity of the hyperbola ${x^2} - {y^2} = 2004$ is

If the line $2x + \sqrt 6 y = 2$ touches the hyperbola ${x^2} - 2{y^2} = 4$, then the point of contact is

The equation of the hyperbola whose vertices are $( \pm 2,0)$ and foci are $( \pm 3,0)$ is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Sum of ${a}^2$ and ${b}^2$ is

$P$ and $Q$ are two points on the rectangular hyperbola $x y=c^2$ such that the abscissa of $P$ and $Q$ are the roots of the equations $x^2-6 x-16=0$. Equation of the chord joining $P$ and $Q$ is

The angle between the asymptotes of the hyperbola $27{x^2} - 9{y^2} = 24$ is

Two tangents are drawn to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ such that the product of their slopes is $c^2$. The locus of their point of intersection is

If the tangent at a point $P$ on $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ cuts one of its directrices at $Q$, then the angle made by $PQ$ at the corresponding focus, is

The eccentricity of the hyperbola $\frac{{\sqrt {1999} }}{3}\left( {{x^2} - {y^2}} \right) = 1,$ is

The circle described on the line joining the foci of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ as a diameter passes through an end of the latus rectum of the parabola $y^2=4 a x$, the length of the latus rectum of the parabola is

If the chords of contact of tangents from two points $\left( {{x_1},\;{y_1}} \right)\mathrm{\;and}\;\left( {{x_2},\;{y_2}} \right)$ to the hyperbola $4{x^2} - 9{y^2} - 36 = 0$ are at right angles, then $\frac{{{x_1}{x_2}}}{{{y_1}{y_2}}}$ is equal to

A hyperbola touches $y$-axis and has its centre and one of the foci are $\left(\frac{5}{2}, 20\right)$ and $(10,24)$ respectively, then length of its transverse axis is.

Let $a$ and $b$ respectively be the semi-transverse and semi-conjugate axes of a hyperbola whose eccentricity $e$ satisfies the equation $9 e^2-18 e+5=0$. If $S(5,0)$ is a focus and $5 x=9$ is the corresponding directrix of hyperbola, then $a^2-b^2$ is equal to

If two perpendicular tangents are drawn from a point $(\alpha, \beta)$ to the hyperbola $x^2-y^2=16$, then the locus of $(\alpha, \beta)$ is

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 $y = 2x + c$ is a tangent to the hyperbola $H,$ Then the value of ${c^2}$ is equal to