Circles
Circle
star_batch_jee_advanced_2025
Grade 11
Question:
If $r_1$ and $r_2$ are the radius of two circles passing through $(-1, 1)$ and touching the lines $x + y = 2, x - y = 2$, and $r_1 + r_2 = a\sqrt{2}$, then $a$ is equal to __________.
Step-by-Step Solution
Key Concept: Setting the center on a given axis and using distance conditions to the line constrains the center location uniquely.
If the circle's center lies on the $x$-axis at $(\lambda, 0)$, then $r^2 = (\lambda + 1)^2 + 1 = \left|\frac{|\lambda - 2|}{\sqrt{2}}\right|^2$. This gives $\lambda^2 + 8\lambda = 0$, so $\lambda = 0$ or $\lambda = -8$. The corresponding circle equations are $x^2 + y^2 = 2$ and $(x+8)^2 + y^2 = 50$. The sum of radii is $r_1 + r_2 = \sqrt{2} + \sqrt{50} = 6\sqrt{2}$.
Correct Answer: 6