Basics of Hyperbola

Point of intersection of the lines $\sqrt{3}x-y-4\sqrt{3}m=0$  and $\sqrt{3}mx+my-4\sqrt{3}=0$ describes

Find centre, directrices, eccentricity, length of conjugate and transverse axis and focii for the hyperbola
$\frac{{\left(3x-4y+1\right)}^2}{100}-\frac{{\displaystyle {\left(4x+3y-9\right)}^2}}{50}=1$

Prove that the square of the length of a focal chord of hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ having slope $m$ is $\frac{4a^2{b}^4{\left(1+m^2\right)}^2}{{\left(a^2{m}^2-b^2\right)}^2}$.

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

The equation $y^2-x^2+2x-1=0$ represents a hyperbola.

Let $\theta \in \left(0,\frac{\pi }{2}\right).$ If the eccentricity of the hyperbola $x^2{\cos }^2\theta -y^2=6{\cos }^2\theta$ is $\sqrt{3}$ times the eccentricity of the ellipse $x^2+y^2{\cos }^2\theta =30{\cos }^2\theta ,$ then $\theta$ is equal to

Find the equation of the hyperbola whose foci are $\left(4,1\right)$ and $\left(8,1\right)$ and whose eccentricity is $2$.

The hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes through the point of intersection of the lines $7x+13y-87=0 \mathrm{and} 5x-8y+7=0$ and its latus rectum is of length $\frac{32\sqrt{2}}{5}$. Eccentricity is $\frac{\sqrt{m}}{5}$. Find $m$.

The length of the latus-rectum of the following hyperbola $9y^2-4x^2=72$ is

Find the length of the chord of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ along the straight line $3x+2y=12$. Also find its mid-point.

Find the equation of the bisector of the obtuse angle between the straight lines $5y-12x=20$ and $3x-4y=8$. Determine which of the angles (acute or obtuse) formed by the lines contains the origin.

The co-ordinates of a point are $P(a \tan (\theta +\alpha ), b \tan (\theta +\beta \left)\right)$, where $\theta$ is a variable; prove that the locus of the point is a hyperbola.

$PQ$ is a double ordinate of the hyperbola $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $O$ is the centre. If $OPQ$ be an equilateral triangle, prove that its eccentricity $e>\frac{2}{\sqrt{3}}$.

Find the eccentricity of the hyperbola $3x^2-y^2=4$.

The transverse and the conjugate axes of a hyperbola are equal. Name the hyperbola and find also its eccentricity.

Examine whether the parametric equation $x=\frac{a}{2}\left(\mu +\frac{1}{\mu }\right), y=\frac{b}{2}\left(\mu -\frac{1}{\mu }\right)$ represents

parabola

hyperbola

ellipse.

Find the equation of the directrices of the hyperbola $12x^2-4y^2=3$.

Find the vertices and the length of transverse and conjugate axes of the hyperbola $4x^2-9y^2=36$.

Find the parametric co-ordinates of the point $\left(\frac{1}{\sqrt{3}}, \frac{1}{2}\right)$ which lies on the hyperbola $12x^2-4y^2=3$.

Does a straight line passing through a point lying inside to the hyperbola always cut hyperbola in two points?