Hyperbola practice · Medium

If the equation of the hyperbola is $\frac{y^2}{9}-\frac{x^2}{27}=1$, then

I. the coordinates of the foci are $(0, \pm 6)$

II. the length of the latus rectum is 18 units

III. the eccentricity is $\frac{4}{5}$

Equations of the asymptotes of the hyperbola $3 x^2+10 x y+8 y^2+14 x+22 y+7=0$ is

The mid point of the chord $4x - 3y = 5$ of the hyperbola $2{x^2} - 3{y^2} = 12$ is

For different values of $\alpha$, the locus of the point of intersection of the two straight lines $\sqrt 3 x - y - 4\sqrt 3 \alpha = 0\;\,\mathrm{and\;}\sqrt 3 \alpha x + \alpha y - 4\sqrt 3 = 0\;$ is

$P$ is a point on the hyperbola $\frac{x^2}{81}-\frac{y^2}{49}=1$. The tangent at $P$ meets the transverse axis at $T, N$ is the foot of the perpendicular from $P$ to the transverse axis. If $O$ is the origin, then $ON.OT$ is equal to.

Any point on the hyperbola $\frac{{{{\left( {x + 1} \right)}^2}}}{{16}} - \frac{{{{\left( {y - 2} \right)}^2}}}{4} = 1$ is of the form

The product of the perpendiculars from the foci on any tangent to the hyperbola $\frac{x^2}{64}-\frac{y^2}{9}=1$ is

The locus of the point of intersection of the lines $x\cot \mathrm{\theta } + y\;\mathrm{cosec \theta } = 2$ and $x\;\mathrm{cosec \theta } + y\cot \mathrm{\theta } = 6$ is

The normal at $P\left(x_1, y_1\right)$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the coordinate axes at $A$ and $B$. If $O$, is the origin and $e$, the eccentricity of the hyperbola, then

Any point on the hyperbola $\frac{{{{\left( {x + 1} \right)}^2}}}{{16}} - \frac{{{{\left( {y - 2} \right)}^2}}}{4} = 1$ is of the form

The condition that the straight line $l x+m y=n$ may be a normal to the hyperbola $b^2 x^2-a^2 y^2=a^2 b^2$ is given by

Length of the straight line $x - 3y = 1$ intercepted by the hyperbola ${x^2} - 4{y^2} = 1$ is

The value of $m$, for which the line $y=m x+\frac{25 \sqrt{3}}{3}$ is a normal to the conic $\frac{x^2}{16}-\frac{y^2}{9}=1$, is

The area of the triangle formed by any tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ with its asymptotes is

A common tangent to $9{x^2} - 16{y^2} = 144$ and ${x^2} + {y^2} = 9$, is

Two conics $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $x^2=-\frac{a}{b} y$ intersect, if

The circle drawn on the line segment joining the foci of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ as diameter cuts the asymptotes at

The locus of point of intersection of straight lines $\frac{x}{a}+\frac{y}{b}=\lambda$ and $\frac{x}{a}-\frac{y}{b}=\frac{1}{\lambda}$ where $\lambda$ is variable, is a

Find the equation of the hyperbola whose foci are $(6,4)$ and $(-4,4)$ and eccentricity is 2 .

The centre of the hyperbola $9 x^2-16 y^2+18 x+32 y-151=0$ is

The product of perpendiculars drawn from any point of a hyperbola to its asymptotes is

Locus of the mid-point of the chords of the hyperbola $x^2-y^2=a^2$, that touch the parabola $y^2=4 a x$, is

The equation of the tangents to the conic $3 x^2-y^2=3$ perpendicular to the line $x+3 y=2$ is

The curve with parametric equation $x = {e^t} + {e^{ - t}}\;y = {e^t} - \;{e^{ - t}}$ and is

If a hyperbola passing through the origin has $3 x+4 y-1=0$ and $4 x-3 y-6=0$ as its assumptions, then the equation of the transverse axis of the hyperbola is

If a variable straight line $x \cos \alpha+y \sin \alpha=p$ which is a chord of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(b>a)$ subtends a right angle at the centre of the hyperbola, then it always touches a fixed circle whose radius, is

The equation of the chord of the hyperbola $25 x^2-16 y^2=400$ that is bisected at point $(5,3)$ is:

If the tangent at the point $P(a \sec \theta, b \tan \theta)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point where a directrix of the hyperbola meets the positive side of the transverse axis, then $\theta$ is equal to
Where $e$ is the eccentricity of the hyperbola.