Question

The normal from origin$\left(O\right)$ to a line meets it at the point $P.$
Let $OP=2$ and $\angle POX=\alpha .$ The line meets the co-ordinate axes at $A$ and $B$ respectively, then the locus of mid point of $AB$ is

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

Answer 1

The correct option is C $x^2+y^2=x^2{y}^2$

Equation of the line $:x\cos \alpha +y\sin \alpha =2$
$\therefore A\equiv (2\sec \alpha ,0),\beta \equiv (0,2\text{cosec}\alpha )$
Let $(h,k)$ be the mid point of $AB$
So, $h=\sec \alpha$ and $k=\text{cosec}\alpha$
$\Rightarrow \frac{1}{h^2}+\frac{1}{k^2}=1$
$\therefore$ Locus is $\frac{1}{x^2}+\frac{1}{k^2}=1$
$x^2+y^2=x^2{y}^2$