Question

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. $30$

Answer 1

Answer: (B)

$AB =\sqrt{26}$
$r ^{2}= CM ^{2}+ AM ^{2}$
$=\left(2 \times \sqrt{\frac{13}{2}}\right)^{2}+\left(\sqrt{\frac{13}{2}}\right)^{2}$
$r ^{2}=\frac{65}{2}$