Hyperbola practice · Mixed difficulty

If the normal at '$\theta$' on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the transverse axis at $G$, and $A$ and $A^{\prime}$ are the vertices of the hyperbola, then $A G \cdot A^{\prime} G=$

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

The locus of the middle points of chords of hyperbola $3 x^2-2 y^2+4 x-6 y=0$ parallel to $y=2 x$ is

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

A common tangent to the conics $x^{2}=6 y$ and $2 x^{2}-4 y^{2}=9$ is:

The foci of a hyperbola coincide with the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$. Find the equation of the hyperbola, if its eccentricity is 2, 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 the tangent and normal to a rectangular hyperbola cut off intercepts $a_1$ and $a_2$ on one axis and $b_1$ and $b_2$ on the other, then

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

If the latus rectum of a hyperbola be 8 and its eccentricity be $3 / \sqrt{5}$, then the equation of the hyperbola is

The one which does not represent a hyperbola is

The mid-point of the chord intercepted by the hyperbola $9{x^2} - 16{y^2} = 144$ on the line $9x - 8y - 10 = 0,$ is

If the eccentricities of the hyperbolas $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $\frac{y^2}{b^2}-\frac{x^2}{a^2}=1$ be $e$ and $e_1$, then find the value of $\frac{1}{e^2}+\frac{1}{e_1^2}$.

The straight line $x + y = \sqrt 2 p$ will touch the hyperbola $4{x^2} - 9{y^2} = 36,$ if

If $2 x-y+1=0$ is a tangent to hyperbola $\frac{x^2}{a^2}+\frac{y^2}{16}=1$, then which of the following are sides of a right angled triangle?

The equation of the hyperbola in the standard form (with transverse axis along the $x$-axis) having the length of the latusrectum $= 9$ unit and eccentricity $= \frac{5}{4}$, is

Let $a>0, b>0$. Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Let $e^{\prime}$ and $l^{\prime}$ respectively be the eccentricity and length of the latus rectum of its conjugate hyperbola. If $\mathrm{e}^2=\frac{11}{14} l$ and $\left(\mathrm{e}^{\prime}\right)^2=\frac{11}{8} l^{\prime}$, then the value of $77 a+44 b$ is equal to :

If $e$ is the eccentricity of the hyperbola $\frac{x^2}{18}-\frac{y^2}{6}=1$ and $r$ is the radius of the circle $x^2 + y^2 - 6x - 18y + 87 = 0,$ then the value of $er$ is equal to:

The point on the curve $3{x^2} - 4{y^2} = 72$, which is nearest to the line $3x + 2y - 1 = 0$, is

Foci of the rectangular hyperbola are ($\pm 7,0$), the equation of the hyperbola is

The locus of the middle points of the portions of the tangents of the hyperbola. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ included between the axes is

Number of points from where perpendicular tangents to the curve $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{{25}} = 1$ can be drawn, is

The reciprocal of the eccentricity of rectangular hyperbola is

If $e_1, e_2$ are the eccentricities of the hyperbola $2 x^2-2 y^2=1$ and the ellipse $x^2+2 y^2=2$ respectively. Then

The length of the transverse axis of a hyperbola is 7 and it passes through the point $(5,-2)$. The equation of the hyperbola is

If the normal at $P$ on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ meets the transverse axis at $G, S$ is a foci and $e$ the eccentricity of the hyperbola then $S G: S P$ is equal to

The locus of middle points of chords of hyperbola $3{x^2} - 2{y^2} + 4x - 6y = 0$ parallel to $y = 2x$ is

The product of the perpendicular from two foci on any tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, is

The equation $a x^2+2 h x y+b y^2+2 g x+2 f y+c=0$ represents a rectangular hyperbola if