Q83 / 150 · MHT CET 2022 (10 Aug Shift 2) · Mathematics

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.

  • A
    $\sqrt{10}$
  • Correct answer
    B
    $3 \sqrt{\frac{5}{2}}$
  • C
    $2 \sqrt{10}$
  • D
    $\frac{3 \sqrt{5}}{2}$

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 $=\frac{1}{2} A C=\frac{1}{2} \sqrt{(6+3)^2+(2-5)^2}=3 \sqrt{\frac{5}{2}}$

Hence, the correct answer is option: 'B'.

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