Consider a circle $C_1:\ x^2+y^2-4x-2y=\alpha-5$. Let its mirror image in the line $y=2x+1$ be another circle $C_2:\ 5x^2+5y^2-10fx-10gy+36=0$. Let $r$ be the radius of $C_2$. Then $\alpha+r$ is equal to ________.
Step-by-Step Solution
Key Concept: $C_1$ has centre $(2,1)$ and radius $\sqrt{\alpha}$. Find the reflection of $(2,1)$ in the line $2x-y+1=0$ to get the centre of $C_2$, then compute $r$.
Centre of $C_2$: $f=-\frac{6}{5},\ g=\frac{13}{5}$. $r=1$. Equal radii $\Rightarrow\alpha=1$. $\alpha+r=2$.
Correct Answer: 2