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10-1.Circle and System of Circles
medium
Let $C$ be a circle passing through the points $A (2,-1)$ and $B (3,4)$. The line segment $AB$ is not a diameter of $C$. If $r$ is the radius of $C$ and its centre lies on the circle $(x-5)^{2}+(y-1)^{2}=\frac{13}{2}$, then $r^{2}$ is equal to
A
$32$
B
$\frac{65}{2}$
C
$\frac{61}{2}$
D
$30$
(JEE MAIN-2022)
Solution
$AB =\sqrt{26}$
$r ^{2}= CM ^{2}+ AM ^{2}$
$=\left(2 \times \sqrt{\frac{13}{2}}\right)^{2}+\left(\sqrt{\frac{13}{2}}\right)^{2}$
$r ^{2}=\frac{65}{2}$
Standard 11
Mathematics
