Hyperbola practice · Mixed difficulty

The line $5x + 12y = 9$ touches the hyperbola ${x^2} - 9{y^2} = 9$ at the point

Let $\mathrm{H}: \frac{-\mathrm{x}^2}{\mathrm{a}^2}+\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4 \sqrt{3}$. Suppose the point $(\alpha, 6), \alpha > 0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$, then $\alpha^2+\beta$ is equal to :

If chords of the hyperbola ${x^2} - {y^2} = {a^2}$ touch the parabola ${y^2} = 4ax.$ Then, the locus of the middle points of these chord is

What is the equation of the chord of hyperbola $25 x^2-16 y^2=400$ whose mid-point is $(5,3)$ ?

If $f(x)=a x^3+b x^2+c x+d,(a, b, c, d$ are rational numbers) and roots of $f(x)=0$ are eccentricities of a parabola and a rectangular hyperbola, then $a+b+c+d$ equals

Coordinates of foci of hyperbola are $\left( { - 5,3} \right)$ and $\left( {7,3} \right)$ and eccentricity is $\frac{3}{2}$. Then ,length of its latusrectum is

Eccentricity of hyperbola whose asymptotes are $3x - 4y = 7$ and $4x + 3y = 8$, is

Consider the hyperbola $H: x^2-y^2=1$ and a circle S with centre $N\left(x_2, 0\right)$. Suppose that $H$ and $S$ touch each other at a point $P\left(x_1, y_1\right)$ with $x_1>1$ and $y_1>0$. The common tangent to $H$ and $S$ at $P$ intersects the $x$-axis at point $M$. If $(l, m)$ is the centroid of triangle $\Delta P M N$, then the incorrect expression, is

Equation $\frac{1}{r} = \frac{1}{8} + \frac{3}{8}\cos \mathrm{\theta }$ represents

If the distance between the directrices of a rectangular hyperbola is 10 unit, then the distance between its foci is

If a hyperbola passes through the point $P\left( {\sqrt 2 ,\sqrt 3 } \right)$ and has foci at $\left( { \pm \,2,0} \right)$, then the tangent to this hyperbola at $P$ also passes through the point

If $x-\sqrt{5} y+c=0$ is a tangent to the hyperbola $\frac{x^2}{25}-\frac{y^2}{4}=1$, then the value of $c$ is

The latus rectum of the hyperbola $16 x^2-9 y^2=144$ is

The difference of the focal distances of any point on the hyperbola is equal to its

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:

The equation of the chord joining two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ on the rectangular hyperbola $x y=c^2$ is

The eccentricity $e$ of the hyperbola $-\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is given by

The line $l x+m y+n=0$ will be a normal to the hyperbola $b^2 x^2-a^2 y^2=a^2 b^2$, if

The equation of the chord of the hyperbola $4 x^2-y^2=$ 4 , which is bisected at the point $(2,-3)$ is

The length of the transverse axis of the rectangular hyperbola $xy = 18$ is

The value of $m$, for which the line $y = mx + 2\;$ becomes a tangent to the conic $4{x^2} - 9{y^2} = 36$ are

If the latusrectum subtends a right angle at the centre of the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1,$ then its eccentricity is

Let the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ be the reciprocal to that of the ellipse $x^2+4 y^2=4$. If the hyperbola passes through a focus of the ellipse, then the equation of the hyperbola, is

Latusrectum of the conic satisfying the differential equation $x d y+y d x=0$ and passing through the point $(2,8)$ is

The equations of the transverse and conjugate axes of a hyperbola are respectively $x+2 y-3=0,2 x-y+4=0$, and their respective lengths are $\sqrt{2}$ and $2 / \sqrt{3}$. The equation of the hyperbola is

The coordinates of a point common to a directrix and an asymptote of the hyperbola $\frac{x^2}{25}-\frac{y^2}{16}=1$ are

The points of the intersection of the curves whose parametric equations are $x=t^2+1, y=2 t$ and $x=2 s, y=\frac{2}{s}$ is given by

A tangent to a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ intercepts a length of unity from each of the coordinate axes, then the point $(a, b)$ lies on the rectangular hyperbola

The equation of the director circle of the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{4} = 1$, is given by

The eccentricity of the curve $x^2-y^2=a^2$ is