Hyperbola practice · Medium

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 $\alpha, \beta$ are the values of $k$ for which the foci of ellipse $\frac{x^2}{k^2 a^2}+\frac{y^2}{a^2}=1$ and hyperbola $\frac{x^2}{a^2}-\frac{y^2}{a^2}=1$ coincide, then the value of $(\alpha+\beta)^2$ is

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

If the tangent and normal to a rectangular hyperbola cut off intercepts $x_1$ and $x_2$ on one axis and $y_1$ and $y_2$ on the other axis, then

If eccentricities of hyperbolas $\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1$ and $\frac{\mathrm{y}^2}{\mathrm{~b}^2}-\frac{\mathrm{x}^2}{\mathrm{a}^2}=1$ are $\mathrm{e}$ and $\mathrm{e} 1$ then $\frac{1}{\mathrm{e}_1^2}+\frac{1}{\mathrm{e}^2}=$

If the tangent at any point $P$ on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ meets the lines $bx - ay = 0$ and $bx + ay = 0$ in the points $Q$ and $R$, then $CQ.CR =$

${P}$ is a variable point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose vertex is ${A}(a, 0)$. The locus of the middle point of $AP$ is

The length of the latusrectum of the hyperbola $xy - 3x - 3y + 7 = 0$ is

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 $(5,3)$ ?

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

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

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

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

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

If the tangent on the point $(2 \sec \phi, 3 \tan \phi)$ of the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$ is parallel to $3 x-y+4=0$, then the value of $\phi$ is

The equation of the hyperbola, referred to its axes as axes of coordinates, given that the distances of one of its vertices from the foci are 9 and 1 units, is

Tangent drawn from the point $(c, d)$ to the hyperbola $9 x^2-64 y^2=576$ make angle $\alpha$ and $\beta$ with the $x$-axis. If $\tan \alpha \tan \beta=1$, then the value of $c^2-d^2$ is equal to

If a rectangular hyperbola $(x-1)(y-2)=4$ cuts a circle $x^2+y^2+2 g x+2 f y+c=0$ at points $(3,4)(5,3)$, $(2,6)$ and $(-1,0)$, then the value of $(g+f)$ is equal to

The length of the semi-transverse axis of the rectangular hyperbola $xy = 32$ is

The asymptote of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ form with any tangent to the hyperbola a triangle whose area is $a^2 \tan \lambda$ in magnitude, then its eccentricity is

If ${e_1}$ and ${e_2}$ are the eccentricities of a hyperbola $3{x^2} - 3{y^2} = 25$ and its conjugate, then

If the normals drawn at four concyclic points on a rectangular hyperbola $x y=c^2$ meet at point $(\alpha, \beta)$ then the centre of the circle has the coordinates

The product of lengths of perpendicular from any point on the hyperbola ${x^2} - {y^2} = 8$ to its asymptotes is

Let the major axis of a standard ellipse equals the transverse axis of a standard hyperbola and their director circles have radius equal to $2 R$ and $R$ respectively. If $e_1$ and $e_2$ are the eccentricities of the ellipse and hyperbola, then the correct relation is

If a circles $x^2+y^2=a^2$ and the rectangular hyperbola $x y=c^2$ intersect in four points, $\left(c t_r, \frac{c}{t_r}\right)$, $r=1,2,3,4$ then $t_1 t_2 t_3 t_4$ is equal to