Coordinate Geometry
Minimum sum of distances between three circles
Grade Class 12
Question:
Two circles $x^2+y^2+14x-6y+40=0$ and $x^2+y^2-2x+6y+7=0$ have centres $C_1$, $C_2$. Another circle with centre $C_3$ on line $3x+4y-16=0$ touches $C_1$ externally and minimises $C_1C_2+C_2C_3+C_3C_1$. Its equation $x^2+y^2+ax+by+c=0$ gives $a+b+c=$
Step-by-Step Solution
Key Concept: Minimum sum of distances $C_1C_2+C_2C_3+C_3C_1$ when $C_1$, $C_3$, image of $C_1$ in the line are collinear.
$a+b+c=0$.
Correct Answer: 2