Question

If a parabola whose length of latus rectum is 4a touches both the coordinate axes then the focus of its locus is

• A. $xy=a^2(x^2+y^2)$
• B. $x^2{y}^2=a^2(x^2+y^2)$
• C. $x^2-y^2=a^2(x^2+y^2)$
• D. $x^2{y}^2=a^2(x^2-y^2)$

Answer 1

The correct option is B $x^2{y}^2=a^2(x^2+y^2)$

Let $(h,k)$ be focus of hyperbola.

Then since it touches both axis, tangent at vertex will be $\frac{x}{a}+\frac{y}{b}=1$ and directrix be $\frac{x}{a}+\frac{y}{b}=0$ and perpendicular distance from directrix would be $a$.

So, Locus $\frac{1}{\sqrt{\frac{1}{h^2}+\frac{1}{K^2}}}=a$

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

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

$⟹x^2{y}^2=a^2(x^2+y^2)$