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10-1.Circle and System of Circles
hard
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
$\frac{585}{66}$
(JEE MAIN-2021)
Solution
Tangent to circle $3 x+4 y=25$
$OP + OQ + OR =25$
In centre $=\left(\frac{\frac{25}{4} \times \frac{25}{3}}{25}, \frac{\frac{25}{4} \times \frac{25}{3}}{25}\right)$
$=\left(\frac{25}{12}, \frac{25}{12}\right)$
$\therefore r ^{2}=2\left(\frac{25}{12}\right)^{2}=2 \times \frac{625}{144}=\frac{625}{72}$
Standard 11
Mathematics
