Hyperbola practice · Easy

If the eccentricity of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\sqrt{5}$ and the distance between the foci is 12, then $b^2-a^2$ is equal to $(3 / 5) k^2$ where $k$ is equal to

The length of the chord of the parabola $y^2=4 a x$, which passes through the vertex and makes an angle $\theta$ with the axis of the parabola, is

The equation of a hyperbola, whose foci are $(5,0)$ and $(-5,0)$ and the length of whose conjugate axis is 8 , is

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

If $e_1$ is the eccentricity of the hyperbola $\frac{x^2}{49}-\frac{y^2}{36}=1$ and $e_2$ is the eccentricity of the hyperbola $\frac{x^2}{36}-\frac{y^2}{49}=1$, then

The normal to the hyperbola $4 x^2-9 y^2=36$ meets the axes in M and ${N}$ and the lines $MP, NP$ are drawn at right angles to the axes. The locus of ${P}$ is the hyperbola

The equation $\frac{{{x^2}}}{{12 - k}} + \frac{{{y^2}}}{{8 - k}} = 1$ represents

Distance between the directrices of the hyperbola $\frac{x^2}{49}-\frac{y^2}{16}=1$ is

The difference in focal distance of any point on the hyperbola $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$ is

If the line $y = 3x + \lambda$ touches the hyperbola $9{x^2} - 5{y^2} = 45,$ then the value of $\lambda$ is

The eccentricity of the hyperbola $9{x^2} - 16{y^2} - 18x - 64y - 199 = 0$ is

The foci of the hyperbola $2{x^2} - 3{y^2} = 5$ are

If the sum of the slopes of the normals from a point $P$ to the hyperbola $x y=c^2$ is equal to $\lambda\left(\lambda \in R^{+}\right)$, then locus of point $P$ is

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

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

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 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