Equation of Normal of Hyperbola in Point ,Parametric Form - Practice Questions & MCQ

Equation of Normal of Hyperbola in Point and Parametric Forms

Point form

The equation of normal at $\left(x_1, y_1\right)$ to the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}=a^2+b^2$.

We know that the equation of tangent in point form at $\left(\mathrm{x}_1, \mathrm{y}_1\right)$

$
\frac{x x_1}{a^2}-\frac{y y_1}{b^2}=1
$
Slope of tangent at $\left(x_1, y_1\right)$ is $\frac{b^2 x_1}{a^2 y_1}$
$\therefore \quad$ Slope of normal at $\left(\mathrm{x}_1, \mathrm{y}_1\right)$ is $-\frac{\mathrm{a}^2 \mathrm{y}_1}{\mathrm{~b}^2 \mathrm{x}_1}$
Hence, the equation of normal at point $\left(\mathrm{x}_1, \mathrm{y}_1\right)$ is

$
\left(\mathrm{y}-\mathrm{y}_1\right)=-\frac{\mathrm{a}^2 \mathrm{y}_1}{\mathrm{~b}^2 \mathrm{x}_1}\left(\mathrm{x}-\mathrm{x}_1\right)
$
$
\text { or } \quad \frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}=a^2+b^2
$
Parametric form
The equation of normal at $(a \sec \theta, b \tan \theta)$ to the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $a x \cos \theta+b y \cot \theta=a^2+b^2$

Proof:
The equation of normal in point form is $\frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}=a^2+b^2$

$
\begin{aligned}
& \text { Put }\left(x_1, y_1\right) \equiv(a \sec \theta, b \tan \theta) \\
& \Rightarrow \quad \frac{a^2 x}{a \sec \theta}+\frac{b^2 y}{b \tan \theta}=a^2+b^2 \\
& \Rightarrow \quad a x \cos \theta+b y \cot \theta=a^2+b^2
\end{aligned}
$

Equation of Normal of Hyperbola in Parametric Form and Slope Form

Parametric form:

The equation of normal at $(a \sec \theta, b \tan \theta)$ to the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $a x \cos \theta+b y \cot \theta=a^2+b^2$

The equation of normal in point form is $\frac{a^2 x}{x_1}+\frac{b^2 y}{y_1}=a^2+b^2{ }_{\text {, put }}\left(x_1, y_1\right) \equiv(a \sec \theta, b \tan \theta)$.

$
\begin{aligned}
& \Rightarrow \quad \frac{a^2 x}{a \sec \theta}+\frac{b^2 y}{b \tan \theta}=a^2+b^2 \\
& \Rightarrow \quad a x \cos \theta+b y \cot \theta=a^2+b^2
\end{aligned}
$
Slope form:

The equation of normal of slope m to the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ are $y=m x \mp \frac{m\left(a^2+b^2\right)}{\sqrt{a^2-m^2 b^2}}$ and coordinate of point of contact is $\left( \pm \frac{a^2}{\sqrt{a^2-m^2 b^2}}, \mp \frac{m b^2}{\sqrt{a^2-m^2 b^2}}\right)$

The equation of normal at $(a \sec \theta, b \tan \theta)$ to the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $a x \cos \theta+b y \cot \theta=a^2+b^2$
$\Rightarrow \quad y=-\frac{a \sin \theta}{b} x+\frac{a^2+b^2}{b \cot \theta}$
Let, $\quad-\frac{\mathrm{a} \sin \theta}{\mathrm{b}}=\mathrm{m}$
$\therefore \quad \sin \theta=-\frac{b m}{a}$
$\therefore \quad \cot \theta= \pm \frac{\sqrt{a^2-b^2 m^2}}{b m}$

Hence, the equation of normal become
$y=m x \mp \frac{m\left(a^2+b^2\right)}{\sqrt{a^2-m^2 b^2}}$, where $m \in\left[-\frac{a}{b}, \frac{a}{b}\right]$

Pair of Tangents

The combined equation of pair of tangents from the point $P\left(x_1, y_1\right)$ to the hyperbola

$
\begin{aligned}
& \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 \text { is }\left(\frac{x^2}{a^2}-\frac{y^2}{b^2}-1\right)\left(\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1\right)=\left(\frac{x x_1}{a^2}-\frac{y y_1}{b^2}-1\right)^2 \\
& \text { or, } \quad S S_1=T^2 \\
& \text { where, } \quad \mathrm{S}=\frac{\mathrm{x}^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}-1 \\
& \mathrm{~S}_1=\frac{\mathrm{x}_1^2}{\mathrm{a}^2}-\frac{\mathrm{y}_1^2}{\mathrm{~b}_2^2}-1 \\
& \mathrm{~T}=\frac{\mathrm{xx}_1}{\mathrm{a}^2}-\frac{\mathrm{yy}_1}{\mathrm{~b}^2}-1
\end{aligned}
$
Note:

The formula $\mathbf{S S}_{\mathbf{1}}=\mathbf{T}^{\mathbf{2}}$ can be used to find the combined equation of pair of tangents for any general hyperbola as well.

Chord of Contact

The equation of chord of contact of tangents from the point $\mathrm{P}\left(\mathrm{x}_1, \mathrm{y}_1\right)$ to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is $\frac{\mathrm{xx}_1}{\mathrm{a}^2}-\frac{\mathrm{yy}_1}{\mathrm{~b}^2}=1$ i.e. $\mathbf{T}=\mathbf{0}$ which is chord of contact $Q R$.

Equation of Chord bisected at a given point:

The equation of chord of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ bisected at a given point $P\left(x_1, y_1\right)$ is $\frac{\mathrm{xx}_1}{\mathrm{a}^2}-\frac{\mathrm{yy}_1}{\mathrm{~b}^2}-1=\frac{\mathrm{x}_1^2}{\mathrm{a}^2}-\frac{\mathrm{y}^2}{\mathrm{~b}^2}-1$
or, $\mathrm{T}=\mathrm{S}_1$

Note:

These formulae can be used for any general hyperbola as well.