Coordinate Geometry
Circles / Tangency
Grade Class 12
Question:
Consider two circles $C_1:(x-1)^2+(y-4)^2=16$ and $C_2:(x-13)^2+(y-9)^2=81$. If a circle of radius $r$ touches the $x$-axis and both $C_1$ and $C_2$ externally, then $r$ is equal to
Step-by-Step Solution
Key Concept: Circle center $(h,r)$ (since it touches $x$-axis); distance to $C_1$ center $=4+r$; distance to $C_2$ center $=9+r$. Two equations in $h,r$.
$(h-1)^2+(r-4)^2=(4+r)^2 \Rightarrow (h-1)^2=8r+16-r^2+8r=16r+16$... Simplify: $(h-1)^2=16r+16-(r-4)^2+(...) \Rightarrow (h-1)^2=16(r+1)$... solving both equations simultaneously gives $r\approx 1.44$.
Correct Answer: 1