Question

The length of the sub-tangent, ordinate of a point and length of the sub-normal at a point (not origin) on $y^2=4ax$ are in

• A. A.P
• B. G.P
• C. H.P
• D. A.G.P

Answer 1

The correct option is B G.P

$y^2=4ax$
At point $P(x_1,y_1)$
$\Rightarrow y_1^2=4a{x}_1$
$\frac{dy}{dx}=\frac{2a}{y}$
Slope of tangent at $P(x_1,y_1)$ is $m=\frac{2a}{y_1}$
Length of sub-tangent $=\left|\frac{y_1}{m}\right|=2x_1$
Length of sub-normal $=|y_{1}m|=2a$
Now, the required sequence is $2x_1,y_1,2a$
Cleary, $y_1^2=4a{x}_1$
Hence, $2x_1,y_1,2a$ are in G.P. with first terms as $\frac{y_1^2}{2a}$ and common ratio as $\frac{2a}{y_1}$.