Mathematics - Circle Question with Solution | TestHub

Equation of $AB$ is $x\cos \alpha +y\sin \alpha =p\Rightarrow \frac{x\cos \alpha }{p}+\frac{y\sin \alpha }{p}=1\Rightarrow \frac{x}{p/\cos \alpha }+\frac{y}{p/\sin \alpha }=1$ So co-ordinates of $A$ and $B$ are $\left(\frac{p}{\cos \alpha },0\right)$ and $\left(0,\frac{p}{\sin \alpha }\right)$; So coordinates of mid point of $AB$ are $\left(\frac{p}{2\cos \alpha },\frac{p}{2\sin \alpha }\right)=\left(x_1,y_1\right)($ let $);x_1=\frac{p}{2\cos \alpha }\&y_1=\frac{p}{2\sin \alpha }$ $\Rightarrow \cos \alpha =p/2x_1$ and $\sin \alpha =p/2y_1;{\cos }^2\alpha +{\sin }^2\alpha =1\Rightarrow \frac{p^2}{4}\left(\frac{1}{x_1^2}+\frac{1}{y_1^2}\right)=1$ Locus of $\left(x_1,y_1\right)$ is $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{p^2}$.