Question

Let the tangent to the circle $x^{2}+y^{2}=25$ at the point $R (3,4)$ meet $x$ -axis and $y$ -axis at point $P$ and $Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having centre at the incentre of the triangle $OPQ$, then $r ^{2}$ is equal to

• A. $\frac{529}{64}$
• B. $\frac{125}{72}$
• C. $\frac{625}{72}$
• D. $\frac{585}{66}$

Answer 1

Answer: (C)

Tangent to circle $3x + 4y = 25$

$OP + OQ + OR =25$
Incentre $=\left(\frac{\frac{25}{4} \times \frac{25}{3}}{25}, \frac{\frac{25}{4} \times \frac{25}{3}}{25}\right)$
$=\left(\frac{25}{12}, \frac{25}{12}\right)$
$\therefore r ^{2}=2\left(\frac{25}{12}\right)^{2}=2 \times \frac{625}{144}=\frac{625}{72}$