A parabola with latus rectum 4a slides such that it touches the positive coordinate axes. Then the locus of its focus is:

Let focus be (h,k)
$\because$  Foot of perpendicular from focus upon tangent lies on tangent at vertex.
$\because$  (h,0) and (0,k) are on tangent at vertex.
$\because$ Equation of tangent at the vertex will be  $\frac{\text{x}}{\text{h}}\text{+}\frac{\text{y}}{\text{k}}\text{=1}$

$\therefore$ perpendicular distance from focus (h,k) upon tangent at the vertex = a
 $\Rightarrow \frac{1+1-1}{\sqrt{\frac{1}{h^2}+\frac{1}{k^2}}}=a$
$\Rightarrow \frac{1}{h^2}+\frac{1}{k^2}=\frac{1}{a^2}$

$\therefore$  Locus is  $\frac{\text{1}}{{\text{x}}^{\text{2}}}\text{+}\frac{\text{1}}{{\text{y}}^{\text{2}}}\text{=}\frac{\text{1}}{{\text{a}}^{\text{2}}}$