Question

The locus of a point which moves such that the sum of the squares of its distances from the three vertices of a triangle is constant, is a circle whose centre is at the.

• A. Incentre of the triangle
• B. Centroid of the triangle
• C. Orthocentre of the triangle
• D. None of these

Answer 1

Answer: (B)

Centroid of the triangle

Answer 2

Let a triangle has its three vertices as (0, 0), (a, 0), (0,b). We have the moving point (h, k) such that $h ^ { 2 } + k ^ { 2 } + ( h - a ) ^ { 2 } + k ^ { 2 } + h ^ { 2 } + ( k - b ) ^ { 2 } = c$

$\Rightarrow 3 h ^ { 2 } + 3 k ^ { 2 } - 2 a h - 2 b k + a ^ { 2 } + b ^ { 2 } = c$

Therefore, $3 x ^ { 2 } + 3 y ^ { 2 } - 2 a x - 2 b y + a ^ { 2 } + b ^ { 2 } = c$

Its centre is $\left( \frac { a } { 3 } , \frac { b } { 3 } \right)$ , which is centroid of $\Delta$ .