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)

$y^{2} = 4ax$
$\Rightarrow 2\,y \frac{dy}{dx} = 4a$
$\Rightarrow \frac{dy}{dx} = \frac{2a}{y}$
Subtangent $= \frac{y}{dy / dx}=\frac{y}{2\, a/y}$
$= \frac{y^{2}}{2a} = \frac{4ax}{2a}= 2x$
Ordinate $= y = \sqrt{4\,ax} = 2\sqrt{a} \sqrt{x}$
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 $G.P$.