Question

The sub normal at $(x,y)$ on the curve $y^2{x}^2=a^2(x^2-a^2)$ varies as.

• A. $x^2$
• B. $\frac{1}{x^2}$
• C. $y^3$
• D. $\frac{1}{x^3}$

Answer 1

The correct option is D $\frac{1}{x^3}$

Given,

$y^2{x}^2=a^2(x^2-a^2)$

$\Rightarrow y^2=a^2(1-\frac{a^2}{x^2})$

$2y\frac{dy}{dx}=a^2(0-(-2a^2{x}^{-3}\left)\right)$

$\Rightarrow \frac{dy}{dx}=\frac{1}{2y}\cdot \frac{2a^4}{x^3}$

$\therefore$ equation of sub normal at $(x,y)=y\cdot \frac{a^4}{x^{3}y}=\frac{a^4}{x^3}$

Varies as $\frac{1}{x^3}$