Question

The locus of a point which moves such that the sum of the square 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)

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 triangle.