Hyperbola - Practice Questions & MCQ

Hyperbola:

A Hyperbola is the set of all points (x, y) in a plane such that the difference of their distances from two fixed points is a constant. Each fixed point is called a focus (plural: foci).

OR,

The locus of a point which moves in a plane such that the ratio of the distance from a fixed point (focus) to the distance from a fixed line (directrix) is constant. The constant is known as eccentricity e and for hyperbola e > 1.

Consider the figure, $O$ is the origin, $S$ and $\mathrm{S}^{\prime}$ are the foci and ZM and $Z^{\prime} \mathrm{M}^{\prime}$ are the directrices.
The foci are $S(a e, 0)$ and $S^{\prime}(a e, 0)$. The equation of directrices: $Z M$ is $x=a / e$ and $Z^{\prime} M^{\prime}$ is $x=a / e$
$P(x, y)$ is any point on the hyperbola and $P M$ is perpendicular to directrix ZM.

$
\begin{aligned}
& \frac{P S}{P M}=e \Rightarrow(P S)^2=e^2(P M)^2 \\
& (xa e)^2+(y0)^2=e^2\left(x\frac{a}{e}\right)^2 \\
& x^2+a^2 e^22 a e x+y^2=e^2 x^22 a e x+a^2 \\
& x^2\left(e^21\right)y^2=a^2\left(e^21\right) \\
& \frac{x^2}{a^2}\frac{y^2}{a^2\left(e^21\right)}=1 \\
& \frac{x^2}{a^2}-\frac{y^2}{b^2}=1, \quad b^2=a^2\left(e^21\right)
\end{aligned}
$
The standard form of the equation of a hyperbola with centre $(0,0)$ and foci lying on the xaxis is

$
\frac{\mathrm{x}^2}{\mathrm{a}^2}\frac{\mathrm{y}^2}{\mathrm{~b}^2}=1 \quad \text { where, } \mathrm{b}^2=\mathrm{a}^2\left(\mathrm{e}^21\right)
$

Important Terms related to Hyperbola:

Centre: All chord passing through point $O$ is bisected at point $O$. Here $O$ is the origin, i.e. $(0,0)$.
Foci: Point $S$ and $S^{\prime}$ is foci of the hyperbola where, $S$ is $(\mathrm{ae}, 0)$ and $S^{\prime}$ is $(\mathrm{ae}, 0)$.
Directrices: The straight line ZM and $Z^{\prime} M^{\prime}$ are two directrices of the hyperbola and their equations are $\mathrm{x}=\mathrm{ae}$ and $\mathrm{x}=\mathrm{ae}$.
Axis: In figure $A A^{\prime}$ is called the transverse axis and the line perpendicular to it through the centre of a hyperbola is called the conjugate axis. $2 a$ is the length of the transverse axis and $2 b$ is the length of the conjugate axis.
Double Ordinate: If a line perpendicular to the transverse axis of the hyperbola meets the curve at Q and $Q^{\prime}$, then $\mathrm{QQ}^{\prime}$ is called double ordinate.
Latus rectum: Double ordinate passing through focus is called latus rectum. Here $L L^{\prime}$ ans $L_1 L_1^{\prime}$ are two latusrectum of a hyperbola.

Eccentricity of Hyperbola: 

Equation of the hyperbola is $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ we have,

$
\begin{aligned}
& \mathrm{b}^2=\mathrm{a}^2\left(\mathrm{e}^21\right) \\
& \mathrm{e}^2=\frac{\mathrm{b}^2+\mathrm{a}^2}{\mathrm{a}^2} \\
& \mathrm{e}=\sqrt{1+\left(\frac{\mathrm{b}^2}{\mathrm{a}^2}\right)} \\
& \mathrm{e}=\sqrt{1+\left(\frac{2 \mathrm{~b}}{2 \mathrm{a}}\right)^2} \\
& \mathrm{e}=\sqrt{1+\left(\frac{\text { conjugate axis }}{\text { transverse axis }}\right)^2}
\end{aligned}
$

Focal Distance of a Point

The difference between the focal distance at any point of the hyperbola is constant and is equal to the length of the transverse axis of the hyperbola.

If $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ is any point on the hyperbola.

$
\begin{aligned}
& \mathrm{SP}=\mathrm{ePM}=e\left(x_1\frac{a}{e}\right)=e x_1a \\
& \mathrm{~S}^{\prime} \mathrm{P}=\mathrm{eP}^{\prime} \mathrm{M}=e\left(x_1+\frac{a}{e}\right)=e x_1+a \\
& \left|\mathrm{~S}^{\prime} \mathrm{P}\mathrm{SP}\right|=\left|\mathrm{ex}_1+\mathrm{a}\mathrm{ex}_1+\mathrm{a}\right|=2 \mathrm{a}
\end{aligned}
$
 

Length of Latusrectum and Parametric Equation of Hyperbola: 

Let Latusrectum $\mathrm{LL}^{\prime}=2 \alpha$
$\mathrm{S}(\mathrm{ae}, 0)$ is focus, then $\mathrm{LS}=\mathrm{SL}^{\prime}=\alpha$
Coordinates of L and $\mathrm{L}^{\prime}$ become (ae, $\alpha$ ) and (ae, $-\alpha$ ) respectively
Equation of hyperbola, $\quad \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$

$
\begin{aligned}
& \therefore \frac{(\mathrm{ae})^2}{\mathrm{a}^2}-\frac{\alpha^2}{\mathrm{~b}^2}=1 \Rightarrow \alpha^2=\mathrm{b}^2\left(\mathrm{e}^2-1\right) \\
& \alpha^2=\mathrm{b}^2\left(\frac{\mathrm{~b}^2}{\mathrm{a}^2}\right) \quad\left[\mathrm{b}^2=\mathrm{a}^2\left(\mathrm{e}^2-1\right)\right] \\
& \alpha=\frac{\mathrm{b}^2}{\mathrm{a}} \\
& \Rightarrow 2 \alpha=\mathrm{LL}^{\prime}=\frac{2 \mathrm{~b}^2}{\mathrm{a}}
\end{aligned}
$

End-points of a latus rectum :

For LR passing through $\mathrm{S}(\mathrm{ae}, 0): L\left(a e, \frac{b^2}{a}\right) ; L^{\prime}\left(a e,-\frac{b^2}{a}\right)$

For $L R$ passing through $S^{\prime}(-a e, 0)$ :

$
L_1\left(-a e, \frac{b^2}{a}\right) ; L_1^{\prime}\left(-a e,-\frac{b^2}{a}\right)
$

Parametric equation of Hyperbola:

The equations $x=a \sec \theta, y=b \tan \theta$ are called the parametric equation of the hyperbola
The circle with center $O(0,0)$ and $O A$ as the radius is called the auxiliary circle of the hyperbola.

Draw PN perpendicular to the x-axis axis and NQ be a tangent to the auxiliary circle. Let be $\angle \mathrm{QON}=\theta$ (This angle is also known as Eccentric Angle). Hence, the parametric equation of circle at point $\mathrm{Q}(\mathrm{a} \cos \theta, \mathrm{a} \sin \theta)$

now, $\mathrm{x}=\frac{\mathrm{ON}}{\mathrm{OQ}} \cdot \mathrm{OQ}=\sec \theta \cdot \mathrm{a}$

$
\begin{aligned}
& \mathrm{x}=\mathrm{a} \sec \theta \\
& \mathrm{P}=(\operatorname{asec} \theta, \mathrm{y})
\end{aligned}
$
P lies on the hyperbola

$
\begin{aligned}
& \frac{a^2 \sec ^2 \theta}{a^2}-\frac{y^2}{b^2}=1 \\
& y= \pm b \tan \theta
\end{aligned}
$
Point P is $(\mathrm{a} \sec \theta, \mathrm{b} \tan \theta)$

Conjugate Hyperbola

Corresponding to every hyperbola, there exists a hyperbola in which the conjugate and the transverse axes of one is equal to the transverse and conjugate axes of the other. Such types of hyperbolas are known as the conjugate hyperbola.

For the hyperbola

$
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1
$
The conjugate hyperbola is

$
\frac{x^2}{a^2}-\frac{y^2}{b^2}=-1
$

Shifted Hyperbola

The standard form of the equation of a hyperbola with center $(h, k)$ and transverse axis parallel to the $x$-axis is

$
\frac{(\mathrm{x}-\mathrm{h})^2}{\mathrm{a}^2}-\frac{(y-k)^2}{b^2}=1
$

or

$
\begin{aligned}
& \frac{(\mathrm{X})^2}{\mathrm{a}^2}-\frac{(Y)^2}{b^2}=1 \\
& \mathrm{x}=\mathrm{X}+\mathrm{h} \quad \text { and } \mathrm{y}=\mathrm{Y}+\mathrm{k}
\end{aligned}
$

Origin is shifted to $(h, k)$.