Hyperbola practice · Easy

Equation of the chord of the hyperbola $25{x^2} - 16{y^2} = 400$ which is bisected at the point $\left( {6,2} \right)$, is

If the eccentricity of a hyperbola is $\sqrt 3$, then the eccentricity of its conjugate hyperbola is

The locus of the foot of the perpendicular drawn from the origin to any tangent to the hyperbola $\frac{x^2}{36}-\frac{y^2}{16}=1$ is

$e_1, e_2$ are respectively the eccentricities of the hyperbola $x^2-y^2 \operatorname{cosec}^2 \theta=5$ and the ellipse $x^2 \operatorname{cosec}^2 \theta+y^2=5$. If $0 < \theta < \pi / 2$ and $e_1=\sqrt{7} e_2$, then $\theta$ is equal to

A hyperbola whose transverse axis is along the major axis of the conic $\frac{x^2}{3}+\frac{y^2}{4}=4$ and has vertices at the foci of the conic. If the eccentricity of the hyperbola is $3 / 2$, then which of the following points does not lie on it?

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

A point on the curve $\frac{x^2}{A^2}-\frac{y^2}{B^2}=1$ is

The equation of the tangent to the hyperbola $4{y^2} = {x^2} - 1$ at the point (1,0), is

If in a hyperbola, the distance between the foci is 10 and the transverse axis has length 8, then the length of its latusrectum is

If the chords of contact of tangents from two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are at right angles, then $\frac{x_1 x_2}{y_1 y_2}$ is equal to

The line $2 x+y=1$ is tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If this line passes through the point of intersection of the nearest directrix and the $x$-axis, then the eccentricity of the hyperbola is

If $e$ and $e_1$ are the eccentricities of a hyperbola and its conjugate, then $\frac{1}{e^2}+\frac{1}{e_1^2}$ is equal to

The tangents to the hyperbola ${x^2} - {y^2} = 3$ are parallel to the straight line $2x + y + 8 = 0$ at the following points:

If $H(x, y)=0$ represents the equation of a hyperbola and $A(x, y)=0, C(x, y)=0$ the joint equation of its asymptotes and the conjugate hyperbola respectively, then for any point $(\alpha, \beta)$ in the plane, $H(\alpha, \beta), A(\alpha, \beta)$, and $C(\alpha, \beta)$ are in

The line $4 \sqrt{2} x-5 y=40$ touches the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ at the point

The area of the triangle formed by the asymptotes and any tangent to the hyperbola $x^2-y^2=a^2$ is

The line $x \cos \alpha+y \sin \alpha=p$ touches the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, if

If the chords of contact of the tangents from two points $\left(x_1, y_1\right)$ and $\left(x_2, y_2\right)$ to the hyperbola $4 x^2-5 y^2=a^2$ are at right angles. Then $16 x_1 x_2+25 y_1 y_2$ is equal to

The equation of the hyperbola referred to its axes as axes of coordinates whose distance between the foci is 16 and eccentricity $\sqrt{2}$, is

The equation to the hyperbola having its eccentricity 2 and the distance between its foci is 8, is

The eccentricity of the hyperbola conjugate to ${x^2} - 3{y^2} = 2x + 8$ is

The curve $x y=c^2$ is said to be

The distance between the foci of the conic $7{x^2} - 9{y^2} = 63$ is equal to

The diameter of $16{x^2} - 9{y^2} = 144$ which is conjugate to $x = 2y$ is

The locus of the equation ${x^2} - {y^2} = 0,$ is

The locus of a point which moves such that the difference of its distances from two fixed points is always a constant, is

If tangent to any member of family of hyperbolas $x y=4 \sin ^2 \theta, \theta \in(0,2 \pi)-\{\pi \}$ is not a normal to any member of family of circles $x^2+y^2-2 x-2 y+\mu=0$, where $\mu$ is any real parameter, then $\theta$ belongs to

The equation of the diameter which is conjugate to $y$ $=3 x$ w.r.t. the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$, is