Circles
Circle
Allen Star Batch
Grade 11
Question:
Let $C_1, C_2, C_3$ be the centres of circles $S_1, S_2, S_3$ respectively, then which of the following must be true:
$C_1C_2 + C_2C_3 = 5$
$C_2C_3 - C_1C_2 = 3$
$C_1C_3 + C_2C_3 = 3$
$C_1C_3 - C_2C_3 = 1$
Step-by-Step Solution
Key Concept: Internally tangent circles have center distances equal to differences of radii; subtracting such relations yields the required relationship.
Given $C_1C_3 = r_3 - r_1$ and $C_1C_2 = r_1 - r_2$, we derive $C_1C_3 - C_1C_2 = r_3 - r_1 - r_1 + r_2 = r_3 - 2r_1 + r_2$. Setting this equal to the given constraint yields $C_1C_3 - C_1C_2 = r_3 - r_1 = 1$.
Correct Answer: 4