Question

A parabola with latus rectum $4 a$ 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. $\frac{1}{x^{2}}+\frac{1}{y^{2}}=\frac{1}{2 a^{2}}, x>0, y>0$
• 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 $\frac{1}{x ^{2}}+\frac{1}{y ^{2}}=\frac{1}{a ^{2}}$