Hyperbola practice · Mixed difficulty

The hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and point $P(4,3)$, then find number of tangents from $P$.

The equation of the tangent to the conic ${x^2} - {y^2} - 8x + 2y + 11 = 0$ at $\left( {2,1} \right)$ is

The equation of a tangent to the hyperbola $16 x^2-25 y^2-96 x+100 y-356=0$, which makes an angle $\pi / 4$ with the transverse axis, is

What is the slope of the tangent drawn to the hyperbola $xy = a,\;\left( {a \ne 0} \right)$ at the point $\left( {a,\;1} \right)?$

If $5{x^2} + \lambda {y^2} = 20$ represents a rectangular hyperbola, then $\lambda$ equals

The difference of the focal distance of any point on the hyperbola $9 x^2-16 y^2=144$ is

The equation $\frac{{{x^2}}}{{2 - \lambda }}--\frac{{{y^2}}}{{\lambda - 5}} - 1 = 0$, represent an ellipse, if

If $m_1$ and $m_2$ are two values of $m$ for which the line $y=m x+2 \sqrt{5}$ is a tangent to the hyperbola $\frac{x^2}{4}-\frac{y^2}{16}=1$, then the value of $\left|m_1+\frac{1}{m_2}\right|$ is equal to

The equation of the hyperbola whose foci are (6, 4) and $( - 4,4)$ and eccentricity 2, is

Tangents at any points on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ cut the axes at $A$ and $B$ respectively. If the rectangle $OAPB,$ where $O$ is the origin is completed, then locus of point $P$ is given by

If the focii of the ellipse $\frac{x^2}{k^2 a^2}+\frac{y^2}{a^2}=1$ and hyperbola $\frac{x^2}{a^2}-\frac{y^2}{a^2}=1$ coincides then $k=$?

Let $F_1, F_2$ are the foci of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $F_3, F_4$ are the foci of its conjugate hyperbola. If $\mathrm{e}_{\mathrm{H}}$ and $\mathrm{e}_{\mathrm{C}}$ are eccentricities of hyperbola and its conjugate hyperbola respectively, then the statement which holds TRUE is :

The pole of the line $l x+m y+n=0$ with respect to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, is

The value of $m$, for which $y=m x+6$ is a tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{49}=1$, is

Equation of the hyperbola whose vertices are $\left( { \pm \,3,0} \right)$ and foci at $\left( { \pm \,5,0} \right)$, is

A line parallel to the straight line $2x - y = 0$ is tangent to the hyperbola $\frac{{{x^2}}}{4} - \frac{{{y^2}}}{2} = 1$ at the point $\left( {{x_1},{y_1}} \right).$ Then $x_1^2 + 5y_1^2$ is equal to

Consider the set of hyperbola $xy = k,k \in R.$ Let ${e_1}$ be the eccentricity when $k = 4$ and ${e_2}$ be the eccentricity when $k = 9$, then ${e_1} - {e_2}$ is equal to

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

The shortest distance between the line $y=x$ and the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$ is

The equation of the transverse and conjugate axis of the hyperbola $16 x^2-y^2+64 x+4 y+44=0$ are

A general point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is

If there are two points $A$ and $B$ on rectangular hyperbola $x y=c^2$ such that abscissa of $A=$ ordinate of $B$, then the locus of point of intersection of tangents at $A$ and $B$ is

If $e$ and ${e_1},$ are the eccentricities of the hyperbolas $xy = {c^2}$ and ${x^2} - {y^2} = {c^2},$, then ${e^2} + e_1^2$ is equal to

If the chords of contacts of the tangents from the points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ to the hyperbola $2 x^2-3 y^2=6$ are at right angle, then $4 x_1 x_2+9 y_1 y_2$ is equal to

The length of the transverse axis of the hyperbola $3 x^2-4 y^2=32$ is

Let the tangent drawn to the parabola $y^2=24 x$ at the point $(\alpha, \beta)$ is perpendicular to the line $2 x+2 y$ $=5$. Then the normal to the hyperbola $\frac{x^2}{\alpha^2}-\frac{y^2}{\beta^2}=1$ at the point $(\alpha+4, \beta+4)$ does NOT pass through the point

If $A$ line with slope $m$ touches the hyperbola $\frac{x^2}{25}-\frac{y^2}{4}=1$ and the parabola $y^2=20 x$. Then the value of $25 m^4-4 m^2$ is equal to

If the hyperbola $\frac{x^2}{b^2}-\frac{y^2}{a^2}=1$ passes through the focus of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, then eccentricity of the hyperbola is

If $\theta$ is an angle between the two asymptotes of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, then $\cos \left(\frac{\theta}{2}\right)$ is equal to

If $x y=\lambda^2-9$ be a rectangular hyperbola whose branches lie only in the second and fourth quadrant, then