Question

Three circles $x^2 + y^2 + 2g_ix + 2f_iy = 0, i=1, 2, 3$ are mutually orthogonal. If $\sum_{i=1}^3 g_i^2 = 4$ and $\sum_{i=1}^3 f_i^2 = 5$, then the equation of the locus of centroid of the triangle formed by joining the centres of these three circles, is.

• A. x^2 + y^2 = 9
• B. x^2 + y^2 = 3
• C. x^2 + y^2 = 2
• D. x^2 + y^2 = 1

Answer 1

Answer: (D)

x^2 + y^2 = 1

Answer 2

The centers of the circles are $C_i = (-g_i, -f_i)$. The centroid $G=(x,y)$ has coordinates $x = -\frac{1}{3}\sum g_i$ and $y = -\frac{1}{3}\sum f_i$. Orthogonality implies $g_ig_j + f_if_j = 0$ for $i \neq j$. Squaring the sum of $g_i$ and $f_i$ and using the given sums of squares and the summed orthogonality conditions leads to $9x^2 = 4 + 2\sum_{i<j}g_ig_j$ and $9y^2 = 5 + 2\sum_{i<j}f_if_j$. Since $\sum_{i<j}g_ig_j + \sum_{i<j}f_if_j = 0$, adding the equations for $9x^2$ and $9y^2$ yields $9(x^2+y^2)=9$, simplifying to $x^2+y^2=1$.