Question

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

• A. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{a^2},x>0,y>0$
• B. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{2}{a^2},x>0,y>0$
• C. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{2a^2},x>0,y>0$
• D. $\frac{1}{x^2}+\frac{1}{y^2}=\frac{4}{a^2},x>0,y>0$

Answer 1

Answer: (A)

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

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