Question

Locus of mid point of the portion between the axes of $x\cos \alpha +y\sin \alpha =p$, where $p$ is constant is

• A. $x^2+y^2=\frac{4}{p^2}$
• B. $x^2+y^2=4p^2$
• C. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{2}{p^2}$
• D. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$

Answer 1

The correct option is D $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$

$x\cos \alpha +y\sin \alpha =p$ touches the coordinate axis at $(\frac{p}{\cos \alpha },0),(0,\frac{p}{\sin \alpha })$

Therefore, the required point $(h,k)$ is $(\frac{P}{2\cos \alpha },\frac{P}{2\sin \alpha })$

$\therefore \cos \alpha =\frac{P}{2h},\sin \alpha =\frac{P}{2k}$

$\Rightarrow {\sin }^2\alpha +{\cos }^2\alpha =\frac{p^2}{4h^2}+\frac{p^2}{4k^2}=1$ $\Rightarrow \frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$