Question

There are two concentric circles \( C_1 \) and \( C_2 \) with radii \( r_1 \) and \( r_2 \). The circles are such that \( C_1 \) fully encloses \( C_2 \). Then what is the radius of \( C_1 \)?
[I.] The difference of their circumference is \( k \) cm.
[II.] The difference of their areas is \( m \) sq. cm.

• A. if the question can be answered with the help of any one statement alone but not by the other statement.
• B. if the question can be answered with the help of either of the statements taken individually.
• C. if the question can be answered with the help of both statements together.
• D. if the question cannot be answered even with the help of both statements together.

Hint:

If one equation gives a difference and another gives a square difference, use algebraic identities.

Answer 1

The Correct Option is C

Let the radii be \( r_1 \) and \( r_2 \), with \( r_1 > r_2 \). From Statement I: \[ 2\pi(r_1 - r_2) = k \Rightarrow r_1 - r_2 = \frac{k}{2\pi} \] From Statement II: \[ \pi(r_1^2 - r_2^2) = m \Rightarrow r_1^2 - r_2^2 = \frac{m}{\pi} \] Now use identity: \[ r_1^2 - r_2^2 = (r_1 - r_2)(r_1 + r_2) \] Using both equations together, we can solve for \( r_1 \). So, both statements are needed.