The orthocentre and centroid of triangle are $A(-3,5)$ and $B(3,3)$ respectively. If $C$ is the circumcenter…
The orthocentre and centroid of triangle are $A(-3,5)$ and $B(3,3)$ respectively. If $C$ is the circumcenter of this triangle, then the radius of the circle having line segment $A C$ as a diameter, is units.
$\sqrt{10}$
$3 \sqrt{\frac{5}{2}}$
$2 \sqrt{10}$
$\frac{3 \sqrt{5}}{2}$
Solution
Circumcentre divides orthocentre $A(-3,0)$ and centroid $B(3,3)$ externally in the ratio 3:1
Hence, $C \equiv\left(\frac{3 \times 3-1 \times(-3)}{3-1}, \frac{3 \times 3-1 \times 5}{3-1}\right) \equiv(6,2)$
Now required radius $=\frac{1}{2} A C=\frac{1}{2} \sqrt{(6+3)^2+(2-5)^2}=3 \sqrt{\frac{5}{2}}$