Circles
Circle
Allen Star Batch
Grade 11
Question:
The centre of a circle $C$ lies on the line $2x - 2y + 9 = 0$ and this circle cuts $x^2 + y^2 = 4$ orthogonally. If this circle passes through two fixed points $(a, b)$ and $(c, d)$, then the value of $a + b + c + d$ is ___.
Step-by-Step Solution
Key Concept: Use the center condition and orthogonality condition simultaneously to determine the circle parameters.
Given circle $x^2 + y^2 + 2gx + 2fy + c = 0$ has center on line $2x - 2y + 9 = 0$, we get $-2g + 2f + 9 = 0$. Since it cuts $x^2 + y^2 - 4 = 0$ orthogonally, we use the orthogonality condition $2gg' + 2ff' = c + c'$ to find $c = 4$. The circle equation becomes $x^2 + y^2 + 9x + 4 + 2f(x + y) = 0$, which passes through intersections of $x^2 + y^2 + 9x + 4 = 0$ and $x + y = 0$, giving points $(- rac{1}{2}, rac{1}{2})$ and $(-4, 4)$.
Correct Answer: 0