First slide

Ellipse

Question

$y = m x + c$ c is a normal to the ellipse
$x^{2} / a^{2} + y^{2} / b^{2} = 1$ if $c^{2}$ is equal to

Moderate

A

$\frac{{(a^{2} - b^{2})}^{2}}{a^{2} m^{2} + b^{2}}$
B

$\frac{{(a^{2} - b^{2})}^{2}}{a^{2} m^{2}}$
C

$\frac{{(a^{2} - b^{2})}^{2} m^{2}}{a^{2} + b^{2} m^{2}}$
D

$\frac{{(a^{2} - b^{2})}^{2} m^{2}}{a^{2} m^{2} + b^{2}}$

Solution

Equation of a normal to the ellipse
$x^{2} / a^{2} + y^{2} / b^{2} = 1$ is
$\frac{a x}{\cos θ} - \frac{b y}{\sin θ} = a^{2} - b^{2}$ , if it represents $m x - y = - c$ then
$\begin{array}{l} \frac{a}{m \cos θ} = \frac{b}{\sin θ} = \frac{- (a^{2} - b^{2})}{c} \\ \Rightarrow \cos θ = \frac{a c}{m (a^{2} - b^{2})}, \sin θ = - \frac{b c}{a^{2} - b^{2}} \\ \Rightarrow \frac{c^{2}}{{(a^{2} - b^{2})}^{2}} [\frac{a^{2}}{m^{2}} + b^{2}] = 1 \\ \Rightarrow \frac{{(a^{2} - b^{2})}^{2} m^{2}}{a^{2} + b^{2} m^{2}} \end{array}$

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