Question

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

• A. $\frac{p}{q}=\frac{8b}{27}$
• B. $\frac{p}{q}=\frac{8b}{(x+a{)}^2}$
• C. $\frac{p}{q}=\frac{(x+a{)}^2}{27}$
• D. $\frac{p}{q}=\frac{8b}{27}(x+a{)}^2$

Answer 1

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

Given equation of curve $b{y}^2={(x+a)}^3$

Let the coordinates of point S on the curve be $(x_1,y_1)$

$\Rightarrow b{y}_1^2={(x_1+a)}^3$

Also, $\frac{dy}{dx}=\frac{3}{2}\frac{{(x+a)}^2}{by}$

Slope of tangent at S$(x_1,y_1)$ is $\frac{3}{2}\frac{{(x_1+a)}^2}{b{y}_1}$

Length of subnormal $SN=|y_{1}m|=\frac{3}{2}.\frac{{(x_1+a)}^2}{b}$

Length of subtangent $ST=\frac{y_1}{m}=\frac{2b{y}_1^2}{3{(x+a)}^2}$

$=\frac{2}{3}(x+a)$

Since, $\frac{p}{q}=\frac{{\left(ST\right)}^2}{SN}$

$\Rightarrow \frac{p}{q}=\frac{\frac{4}{9}(x+a{)}^2}{\frac{3}{2}(x+a{)}^2}$

$\Rightarrow \frac{p}{q}=\frac{8b}{27}$