Question

If a rectangular hyperbola of latus rectum $4$ units passing through $(0,0)$ have $(2,0)$ as its one focus, then equation of locus of the other focus is

• A. $x^2+y^2=36$
• B. $x^2+y^2=4$
• C. $x^2-y^2=4$
• D. $x^2-y^2=36$

Answer 1

The correct option is A $x^2+y^2=36$

For a rectangular hyperbola, length of latus rectum $=\frac{2b^2}{a}=2a=2b(\because a=b)$
So, $2a=4$
Given focus is $S(2,0)$
Hyperbola passes through the point $P(0,0)$
Let $S^{\prime }(h,k)$ be other focal point.
Now, $|S^{\prime }P-SP|=2a$
$\Rightarrow |\sqrt{h^2+k^2}-2|=4$ $\Rightarrow \sqrt{h^2+k^2}=6\Rightarrow h^2+k^2=36$
So, Locus of $(h,k)$ is $x^2+y^2=36$