Basics of Hyperbola

Let $S$ be the focus of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ lying on the positive $X-$ axis and $P\left(5,y_1\right)$ be point on the hyperbola. Then $\mathrm{SP}=$

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

The equation of the conic with focus at $(1,\mathrm{ }-1)$ , directrix along $\mathrm{x}-\mathrm{y}+1=\mathrm{0 }$ and with eccentricity $\sqrt{2}$ is

Find the foci of the hyperbola $2x^2-3y^2=5$.

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.

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.

Determine whether the point $\left(2, 1\right)$ lies outside, upon or inside the hyperbola $x^2-y^2=4$.

If $C$ be the centre and $S$, $S’$ are the foci of the rectangular hyperbola $x^2-y^2=a^2$, prove that for any point $P$ on the hyperbola, $\mathrm{SP}\cdot {\mathrm{S}}^{\mathrm{\prime }}\mathrm{P}={\mathrm{CP}}^2$.

A point moves on a plane in such a way that the difference of its distances from the points ($6, 0$) and ($-6, 0$) is always equal to $8$ unit. Prove that its locus is a hyperbola and find its equation.

The equation of a directrix is $x-y+3=0$, co ordinate of a focus are ($-1, 1$) and eccentricity $3$ for a hyperbola. Find the equation of the hyperbola.
 

For a hyperbola, the coordinates of a focus are ($1, 1$), the equation of a directrix is $3x+4y+8=0$ and eccentricity is $2$. Find the equation of the hyperbola.
 

The coordinates of a focus of a hyperbola are ($2, -1$), the equation of a directrix is $2x+3y=1$ and the eccentricity is $2$. Find the equation of the hyperbola.
 

Find the equation of the hyperbola whose focus is at ($0, 0$), eccentricity is $2$ and the equation of its directrix is $3x+4y+1=0$.

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

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 a

(i) parabola

(ii) hyperbola

(iii) ellipse.

For the hyperbola $\frac{{\mathrm{x}}^2}{{\mathrm{a}}^2}-\frac{{\mathrm{y}}^2}{{\mathrm{b}}^2}=1,$ distance between the foci is $10$ units. From the point $\left(2,\sqrt{3}\right),$ perpendicular tangents are drawn to the hyperbola, then the value of $\left|\frac{\mathrm{b}}{\mathrm{a}}\right|$ is

If $P(\sqrt{2}\sec \theta ,\sqrt{2}\tan \theta )$ is a point on the hyperbola whose distance from the origin is $\sqrt{6}$ where $P$ is in the first quadrant then $\theta =$

Shortest distance between the two curves $y=e^x$ and $y=\ln x$ is

The curve represented by $4x^2-4y^2-6xy+5x+5y=7$ is

A hyperbola has its centre at $(0, 0)$, passes through the point $(4, 2)$ and has transverse axis of length $4$ along the $\mathrm{x}\text{-axis.}$ Then the eccentricity of hyperbola is: