Question

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

Answer 1

Step-by-step explanation:

Answer 2

The focus is $x^2y^2=a^2(y^2+x^2)$.

Step-by-step explanation:  

Given:

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,

$\frac{1}{\sqrt{\frac{1}{h^2}+\frac{1}{k^2}}} =a$

$\sqrt{\frac{1}{h^2}+\frac{1}{k^2}} =\frac{1}{a}$

${\frac{1}{h^2}+\frac{1}{k^2} =\frac{1}{a^2}$

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

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

$a^2(y^2+x^2)=x^2y^2$

$x^2y^2=a^2(y^2+x^2)$