Question

For the curve $b{y}^2=(x+a{)}^3$ the square of subtangent is proportional to

• A. $(Subnormal{)}^{1/2}$
• B. Subnormal
• C. $(Subnormal{)}^{3/2}$
• D. None of these

Answer 1

The correct option is B Subnormal

$b{y}^2=(x+a{)}^3\Rightarrow 2by.\frac{dy}{dx}=3(x+a{)}^2\Rightarrow \frac{dy}{dx}=\frac{3}{2by}(x+a{)}^2$
$\therefore$ Subnormal $=y\frac{dy}{dx}=\frac{3}{2b}(x+a{)}^2$
$\therefore$ Subtangent =$\frac{y}{\left(\frac{dy}{dx}\right)}=\frac{y}{\frac{3(x+a{)}^2}{2by}}=\frac{2b{y}^2}{3(x+a{)}^2}$
$=\frac{2b\frac{(x+a{)}^3}{b}}{3(x+a{)}^2}=\frac{2}{3}(x+a)$
$\therefore (Subtangent{)}^2=\frac{4}{9}(x+a{)}^2$
and $\frac{(Subtangent{)}^2}{Subnormal}=\frac{\frac{4}{9}(x+a{)}^2}{\frac{3}{2b}(x+a{)}^2}=\frac{8b}{27}$
$\Rightarrow (Subtangent{)}^2=constant\times (Subnormal).\therefore (Subtangent{)}^2\propto \left(Subnormal\right)$