MHT CET · Maths · Straight Lines
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 \(\sqrt{10}\)
- B \(3 \sqrt{\frac{5}{2}}\)
- C \(2 \sqrt{10}\)
- D \(\frac{3 \sqrt{5}}{2}\)
Answer & Solution
Correct Answer
(B) \(3 \sqrt{\frac{5}{2}}\)
Step-by-step Solution
Detailed explanation
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, \(C \equiv\left(\frac{3 \times 3-1 \times(-3)}{3-1}, \frac{3 \times 3-1 \times 5}{3-1}\right) \equiv(6,2)\)
Now required radius \(=\frac{1}{2} A C=\frac{1}{2} \sqrt{(6+3)^2+(2-5)^2}=3 \sqrt{\frac{5}{2}}\)
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