Question

The tangent at any point to the circle $x^2+y^2=r^2$ meets the coordinate axes at $A$ and $B$. If lines drawn parallel to the coordinate axes through $A$ and $B$ intersect at $P$, then the locus of $P$ is

• A. $x^2+y^2=r^2$
• B. $x^2{y}^2-r^2(x^2+y^2)=0$
• C. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{r^2}$
• D. $\frac{1}{x^2}+\frac{1}{y^2}=r^2$

Answer 1

Answer: (C)

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

Equation of a tangent is $x\cos \theta +y\sin \theta =r$
$A(\frac{r}{\cos \theta },0)$, $B(0,\frac{r}{\sin \theta })$, $P(\frac{r}{\cos \theta },\frac{r}{\sin \theta })$
$=P(x,y)$
$\Rightarrow \frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{r^2}$