Hyperbola practice · Mixed difficulty

$P$ 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.

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 latusrectum is $\sqrt{14}$. Then the square of the eccentricity of $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is:

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 eccentricity of the conic represented by $x^2-y^2-4 x+4 y+16=0$, is

The locus of the poles of the chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ which subtend a right angle at its centre is

Eccentricity of the hyperbola $2 x^2-2 y^2-3 x y+9 x+12 y=0$ is equal to

The equation of the normal to the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$ at $\left( { - 4,0} \right)$ 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?

What is the slope of the tangent line drawn to the hyperbola $x y=a(a \neq 0)$ at the point $(a, 1)$ ?

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 locus of a point $\mathrm{P}(\alpha, \beta)$ which is 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

The locus of the point of intersection of the tangents to the hyperbola $\frac{x^2}{16}-\frac{y^2}{36}=1$ which are at right angles is

The centre of the hyperbola $9 x^2-36 x-16 y^2+96 y-252=0$, is

Acceleration of a particle moving along a straight line is a function of velocity as $a=2 \sqrt{v}$. At $t=2 s$, its velocity $v=16 \mathrm{~ms}^{-1}$. Its velocity at $t=3$ s will be

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 co-ordinate of focus of $9 x^2-16 y^2+18 x+32 y-151=0$

If the line $l x+m y+n=0$ touches the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then

The locus of the point of intersection of the tangents at the end-points of normal chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, is

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

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 the hyperbola is:

The coordinates of the centre of the hyperbola ${x^2} + 3xy + 2{y^2} + 2x + 3y + 2 = 0$

Equation of the chord of the hyperbola $25{x^2} - 16{y^2} = 400$ which is bisected at the point $\left( {6,2} \right)$, is

If the eccentricity of a hyperbola is $\sqrt 3$, then the eccentricity of its conjugate hyperbola is

The locus of the foot of the perpendicular drawn from the origin to any tangent to the hyperbola $\frac{x^2}{36}-\frac{y^2}{16}=1$ is

$e_1, e_2$ are respectively the eccentricities of the hyperbola $x^2-y^2 \operatorname{cosec}^2 \theta=5$ and the ellipse $x^2 \operatorname{cosec}^2 \theta+y^2=5$. If $0 < \theta < \pi / 2$ and $e_1=\sqrt{7} e_2$, then $\theta$ is equal to

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

A hyperbola whose transverse axis is along the major axis of the conic $\frac{x^2}{3}+\frac{y^2}{4}=4$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $3 / 2$, then which of the following points does not lie on it?