Question

The locus of the mid-point of the intercept of the variable line $x\cos \alpha +y\sin \alpha =p$ ($p$ is constant) between the coordinates axes is

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

Answer 1

Answer: (C)

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

We have,

$x\cos \alpha +y\sin \alpha =p$..........(1)

Then, the solution is shown below,

This line intercept the axes.

Thus, the co-ordinate of the point where the line intercept

$x-axis$ is,

$(\frac{P}{\cos \alpha },0)$

and, the co-ordinate of the point where the line intercept y-axis is

$(0,\frac{P}{\sin \alpha })$

The mid-point $R$ of the line is given by,

$R(h,k)=(\frac{\frac{P}{\cos \alpha }+0}{2},\frac{0+\frac{P}{\sin \alpha }}{2})$

$=(\frac{P}{2\cos \alpha },\frac{P}{2\sin \alpha })$

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

Elementary the sine and cosine terms, we get.

${\cos }^2\alpha +{\sin }^2\alpha =1$

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

$\Rightarrow P^2(h^2+k^2)=4h^2{k}^2$

Thus, the locus is given by

$p^2(x^2+y^2)=4x^2{y}^2$

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

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