Circles
Circle
star_batch_jee_advanced_2025
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:
C1C2 + C2C3 = 5
C2C3 - C1C2 = 3
C1C3 + C2C3 = 3
C1C3 - C2C3 = 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.
Step 1: Identify the given relations between the distances between centers and the radii of the circles.
The problem provides the following relationships:
$$ C_1C_3 = r_3 - r_1 $$
$$ C_1C_2 = r_1 - r_2 $$
Step 2: Compute the difference $C_1C_3 - C_1C_2$ using the given relations.
Substitute the expressions from Step 1 into the difference:
$$ C_1C_3 - C_1C_2 = (r_3 - r_1) - (r_1 - r_2) $$
$$ C_1C_3 - C_1C_2 = r_3 - r_1 - r_1 + r_2 $$
$$ C_1C_3 - C_1C_2 = r_3 - 2r_1 + r_2 $$
Step 3: Apply the given constraint as stated in the original solution.
The original solution states that by setting this expression equal to a given constraint, it yields:
$$ C_1C_3 - C_1C_2 = r_3 - r_1 = 1 $$
This implies two conditions:
1. The expression $r_3 - 2r_1 + r_2$ is equal to $r_3 - r_1$. This holds if and only if $r_2 = r_1$.
2. The value of $r_3 - r_1$ (and thus $C_1C_3 - C_1C_2$) is $1$.
Step 4: State the final result derived from the solution.
Based on the explicit statement in the original solution, the final result is:
$$ C_1C_3 - C_1C_2 = 1 $$
The expression derived from the provided solution is $C_1C_3 - C_1C_2 = 1$. This expression does not directly match any of the given options. The options involve $C_2C_3$, which was not part of the derivation in the original solution provided. Therefore, based on the provided solution, a direct match to the options cannot be made.
Correct Answer: 4