Question

Let the locus of the centre $(\alpha, \beta), \beta>0$, of the circle which touches the circle $x ^2+( y -1)^2=1$ externally and also touches the $x$-axis be $L$. Then the area bounded by $L$ and the line $y=4$ is :

• A. $\frac{32 \sqrt{2}}{3}$
• B. $\frac{40 \sqrt{2}}{3}$
• C. $\frac{64}{3}$
• D. $\frac{32}{3}$

Answer 1

Answer: (C)

$ (\alpha-0)^2+(\beta-1)^2=(\beta+1)^2$
$\alpha^2=4 \beta $
$ x^2=4 y$
$ A=2 \int\limits_0^4\left(4-\frac{x^2}{4}\right) d x=\frac{64}{3}$