Circles
Three circles between parallel lines — Descartes theorem
MJAT_TS6_P2
Grade 12
Question:
Lines $L_1$, $L_2$ are parallel. $C_1$ is tangent to both. $C_2$ is tangent to $C_1$ externally and $L_1$; $C_3$ is tangent to $C_1$ externally and $L_2$; both between the lines and tangent to each other. Radii $r_2=5\sqrt{2}$, $r_3=10\sqrt{2}$. Diameter of $C_1$:
A) $15\sqrt{2}$
B) $15$
C) $20$
D) $40$
Step-by-Step Solution
Key Concept: Let $r$ = radius of $C_1$. Distance from $L_1$ to $L_2=2r$. For $C_2$ (tangent to $L_1$ and $C_1$): $x^2=4rr_1$ (horizontal distance). For $C_3$: $p^2=4rr_2$. For $C_2,C_3$ tangent: $(x+p)^2+(r_2-r_1)^2=(r_1+r_2)^2$.
Diameter $=\mathbf{40}$. Answer: **D**.
Correct Answer: D