Question

Lengthof the subtangent at $(a, a)$ on the curve $y^2 = \frac{x^2}{2a +x}$ is equal to

• A. $\frac{18}{5}$
• B. $\frac{18a}{5}$
• C. $\frac{18a^2}{5}$
• D. $ - \frac{18a^2}{5}$

Answer 1

The Correct Option is C

Given, $y^{2} = \frac{x^{2}}{2a+x} $
$\Rightarrow 2y \frac{dy}{dx} = \frac{\left(2a+x\right)\left(2x\right)-x^{2} \left(1\right)}{\left(2a+x^{2}\right)}$
$ \Rightarrow \left[\frac{dy}{dx}\right]_{\left(a,a\right)} = \frac{\left(2a+a\right)\left(2a\right)-a^{2}}{2\times a\left(2a+a\right)^{2}}$
$ = \frac{\left(3a\right)\left(2a\right)-a^{2}}{2a\left(3a\right)^{2}} $
$= \frac{6a^{2} -a^{2}}{2a \times9a^{2}} = \frac{5a^{2}}{18a^{3}} $
length of subtangent at (a, a)
$= \frac{y}{\left(\frac{dy}{dx}\right)_{\left(a,a\right)}}= \frac{a}{\frac{5}{18a}} = a\times \frac{18a}{5}=\frac{18a^{2}}{5} $