Hyperbola practice · Mixed difficulty

If $\frac{x^2}{a+7}+\frac{y^2}{5-a}=1$ represents a hyperbola, then

The eccentricity of the hyperbola $\frac{\sqrt{1999}}{3}\left(x^2-y^2\right)=1$ is

If the normal at $P$ to the rectangular hyperbola $x^2-y^2=4$ meets the axes of $x$ and $y$ in $G$ and $g$ respectively and $C$ is the centre of the hyperbola, then $2 P C=$

If the origin is shifted to the point $\left(3, \frac{7}{2}\right)$ and the axes are rotated through an angle $\theta$ in a clockwise direction, so that the equation of the given hyperbola changes to the standard form $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then $\theta$ is

If $\alpha+\beta=3 \pi$, then the chord joining the points $\alpha$ and $\beta$ for the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through

If the chord of contact of tangents from a point $P$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends a right angle at the centre, then the locus of ${P}$ is

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ does not have mutually $\perp r$ tangents, then possible values of eccentricities are
a) $\frac{3}{2}\qquad$  b) $\frac{8}{7}\qquad$ c) $\frac{5}{3}\qquad$ d) $\frac{7}{5}$

Find the equation of the hyperbola whose directrix is $2 x+y=1$, focus is $(1,1)$ and eccentricity is $\sqrt{3}$.

The eccentricity of the hyperbola $4 x^2-9 y^2=16$ is

If the eccentricity of the hyperbola $x^2-y^2 \sec ^2 \alpha=5$ is $\sqrt{3}$ times the eccentricity of the ellipse $x^2 \sec ^2 \alpha+y^2=25$, then a value of $\alpha$ is

Consider a branch of the hyperbola $x^2-2 y^2-2 \sqrt{2} x-4 \sqrt{2} y-6=0$ with vertex at the point $A$. Let $B$ be one of the end points of its latusrectum. If $\mathrm{C}$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $A B C$, is

The locus of a point $P\left( {\mathrm{\alpha },\mathrm{\beta }} \right)$ moving under the condition that the line $y = \mathrm{\alpha }x + \mathrm{\beta }$ is a tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$, is

The condition that a straight line with slope $m$ will be normal to parabola $y^2=4 a x$ as well as a tangent to rectangular hyperbola $x^2-y^2=a^2$ is

The equation of the normal at the point $(a\sec \mathrm{\theta },\;b\tan \mathrm{\theta })\;$ of the curve $\;{b^2}{x^2} - {a^2}{y^2} = {a^2}{b^2}\;$ is

Tangent to hyperbola $x y=c^2$ at point $P$ intersects the $x$-axis at $T$ and the $y$-axis at $T^{\prime}$. Normal to hyperbola at ${P}$ intersects the ${x}$-axis at ${N}$ and the ${y}$-axis at ${N}^{\prime}$. If the area of the triangles $PNT$ and $PN^{\prime}T^{\prime}$ are $\Delta$ and $\Delta^{\prime}$ respectively then $\frac{1}{\Delta}+\frac{1}{\Delta^{\prime}}$, is equal to

The eccentricity of the curve $x^2-y^2=1$ is

The equation $x^2+4 x y+y^2+2 x+4 y+2=0$ represents

The eccentricity of the conic defined by $\left|\sqrt{(x-1)^2+(y-2)^2}-\sqrt{(x-5)^2+(y-5)^2}\right|=3$ is $\mathrm{K}$, then $3 \mathrm{~K}=$

If the slope of a tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $2 \sqrt{2}$, then the eccentricity $e$ of the hyperbola lies in the interval.

From a point on the line $y=x+c, c$ (parameter), tangents are drawn to the hyperbola $\frac{x^2}{2}-\frac{y^2}{1}=1$ such that chords of contact pass through a fixed point $\left(x_1, y_1\right)$. Then, $\frac{x_1}{y_1}$ is equal to

The eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{25}=1$ is

If the circle ${x^2} + {y^2} = {a^2}$ intersects the hyperbola $xy = {c^2}$ in four points $P\left( {{x_1},{y_1}} \right),Q\left( {{x_2},{y_2}} \right),R\left( {{x_3},{y_3}} \right)$ and $S\left( {{x_4},{y_4}} \right)$, then

A common tangent to $x^2-2 y^2=18$ and $x^2+y^2=9$ is

The latusractum of the hyperbola $9{x^2} - 16{y^2} + 72x - 32 - 16 = 0$ is

If the angle between the asymptotes of the hyperbola $\left(\frac{x^2}{a^2}-\frac{y^2}{b^2}\right)$ is $2 \theta$ and $e$ is its eccentricity then $\sin \theta$ is equal to

Let $P(a\sec \mathrm{\theta },b\tan \mathrm{\theta })$ and $Q(a\sec \phi ,b\tan \phi ),$ where $\mathrm{\theta } + \phi = \frac{\mathrm{\pi }}{2}$ be two points on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1.$. If $\left( {h,k} \right)$ is the point of intersection of normals at $P$ and $Q$, then $k$ is equal to

$P$ is any point on the hyperbola $x^2-y^2=a^2$. If $F_1$ and $F_2$ are the foci of the hyperbola and $P F_1 \cdot P F_2=\lambda(O P)^2$, where $O$ is the origin, then $\lambda$ is equal to

If a hyperbola passes through the point $P(10,\,16)$ and it has vertices at $( \pm 6,\,0),$ then the equation of the normal to it at $P$ is

The equation of the tangent to the hyperbola $16 x^2-9 y^2=144$ at $(5,8 / 3)$, is

If the normal at the point '$t_1$' to the rectangular hyperbola $x y=c^2$ meets it again at the point '$t_2$', then