Question

A tangent to the circle $x^2+y^2=a^2$ intersects the coordinate axes at $A$and $B$. The locus of the point of intersection of the lines passing through $A$ and $B$ and parallel to the coordinate axes is

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

Answer 1

The correct option is A $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{a^2}$

At $(a\cos \theta ,\sin \theta )$ $e{q}^n$ of tanget to $x^2+y^2=a^2$ is $x\cos \theta +y\sin \theta =a$

$\therefore$ Line passing through A and parallel to Y-axis.

$X=\frac{a}{\cos \theta }$

Line passing through B and parallel to X-axis.

$Y=\frac{a}{\sin \theta }$

$\therefore$ locus of point $(\frac{a}{\cos \theta },\frac{a}{\sin \theta })$

$x=\frac{a}{\sin \theta }$

$\Rightarrow 1=\frac{a^2}{x^2}+\frac{a^2}{y^2}$