Question

Locus of the middle point of the portion of the line $x\cos \theta +y\sin \theta =p$ which is intercepted between the axes given that $p$ is constant, is

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

Answer 1

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

x-intercept of line$=2h$

$x=\frac{p}{\cos \theta }=2h$

$\cos \theta =\frac{p}{2h}$

y-intercept of line$=2k$

$y=\frac{p}{\sin \theta }=2k$

$\sin \theta =\frac{p}{2k}$

Also, ${\sin }^2\theta +{\cos }^2\theta =1$

$\frac{p^2}{4k^2}+\frac{p^2}{4h^2}=1$

$\frac{p^2}{4x^2}+\frac{p^2}{4y^2}=1$

$\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$.