Question

A right angled triangle ABC right angled at A and $b=18cm$ and $c=63cm$. The circle $C_1$ is drawn with centre lying on hypotenuse BC and touching the side AB and CA. Another circle $C_2$ is drawn concentric with $C_1$ and passing through A. The area of the circle $C_2$ is

• A. $144\pi c{m}^2$
• B. $196\pi c{m}^2$
• C. $288\pi c{m}^2$
• D. $392\pi c{m}^2$

Answer 1

The correct option is D $392\pi c{m}^2$

D is equidistant from AB & AC
$\therefore$ it lies on angle bisector.
Equation of $BC=y=\frac{x}{63}+\frac{y}{18}=1$
Equation of $AD=y=x$
The coordinates of D are $(14,14)$
$r=AD=\frac{2bc\cos \frac{A}{2}}{b+c}=\frac{2\times 18\times 63}{81}\times \frac{1}{\sqrt{2}}=14\sqrt{2}$
Area $=\pi \times (14\sqrt{2}{)}^2=392\pi c{m}^2$