Question

If at any point S of the curve $b{y}^2={(x+a)}^3$ the relation between subnormal SN and subtangent ST be $p\left(SN\right)=q{\left(ST\right)}^2$ then $\frac{p}{q}$ is equal to -

• A. $\frac{8b}{27}$
• B. $\frac{8a}{27}$
• C. $\frac{b}{a}$
• D. None of these

Answer 1

The correct option is A $\frac{8b}{27}$

$b{y}^2={(x+a)}^3$
$\Rightarrow 2by\frac{dy}{dx}=3{(x+a)}^2$
$\Rightarrow \frac{dy}{dx}=\frac{3{(x+a)}^2}{2by}$
$\therefore$ S.N. $=y\times \frac{dy}{dx}=\frac{3}{2}\frac{{(x+a)}^2}{b}$
and S.T. $=y\frac{dx}{dy}=\frac{2b{y}^2}{3{(x+a)}^2}$
$\therefore \frac{p}{q}=\frac{{\left(ST\right)}^2}{SN}$
$\Rightarrow \frac{p}{q}=\frac{8b{\left\{{(x+a)}^3\right\}}^2}{27{(x+a)}^6}$
$\Rightarrow \frac{p}{q}=\frac{8b}{27}$
Hence, option 'A' is correct.