Question

The subtangent, ordinate and subnormal to the parabola $y^2=4ax$ at a point (different from the Origin) are in

• A. G.P
• B. A.P.
• C. H.P
• D. none of these

Answer 1

Answer: (A)

$y^2=4ax$
$\Rightarrow 2y\frac{dy}{dx}=4a$
$\Rightarrow \frac{dy}{dx}=\frac{2a}{y}$
Subtangent $=\frac{y}{dy/dx}=\frac{y}{2a/y}$
$=\frac{y^2}{2a}=\frac{4ax}{2a}=2x$
Ordinate $=y=\sqrt{4ax}=2\sqrt{a}\sqrt{x}$
Subnormal $=y\frac{dy}{dx}=2a$
Since (subtangent) (subnormal) $=\left(2x\right)\left(2a\right)$
$=4ax=$ (ordinate)${}^2$
$\Rightarrow$ subtangent, ordinate and subnormal are in $G.P$.