Question

If parabola with focus $\left(\frac{2}{5}, \frac{4}{5}\right)$ touches $x$ and $y$-axis at $A$ and $B$ then
Area of $\triangle OAB$, where $O$ is origin, is equal to

• A. $1$
• B. $\frac{4}{25}$
• C. $\frac{3}{5}$
• D. 2

Answer 1

Answer: (A)

If parabola touches $x$-axis at $A(a, 0)$ and $y$-axis at $B(0, b)$ then focus is point of intersection of circles with diameter $OA$ and $OB$.
Equation of circle with $OA$ as diameter is
$x(x-a)+y^2=0 $
$\Rightarrow x^2+y^2-a x=0$.....(1)
and equation circle with $OB$ as diameter :
$x^2+y^2-b y=0$ .....(2)
for point of int. of (1) and (2) ax - by $=0 \Rightarrow y=\frac{a}{b} x$
from(1) $ x^2+\frac{a^2}{b^2} x^2-a x=0 \Rightarrow x=0$ or $x=\frac{a b^2}{a^2+b^2}=\left(\frac{2}{5}, \frac{4}{5}\right)$ $\therefore a =1, b =2$
$\therefore$ Area of $\triangle OAB =\frac{1}{2} \times 1 \times 2=1$