Question

Three circles A, B and C have a common centre O. A is the inner circle, B is the middle circle, and C is the outer circle. P is a point on the outer circle C, and the radius OP cuts the inner circle at X and the middle circle at Y, such that \(OX = XY = YP\). The ratio of the area of the region between the inner and middle circles to the area of the region between the middle and outer circles is:

• A. \(\dfrac{1}{3}\)
• B. \(\dfrac{2}{5}\)
• C. \(\dfrac{3}{5}\)
• D. \(\dfrac{1}{5}\)

Hint:

Since OX = XY = YP, the three circles have radii in the ratio 1:2:3; use area = pi times radius squared for each ring and take the ratio.

Answer 1

The Correct Option is C

Step 1: Assign a common unit length using the equal segments.
Let \(OX = XY = YP = r\). Since O, X, Y and P all lie on the same radius line OP of the outer circle, the distances from the centre O build up in steps of \(r\) along this line.

Step 2: Find the radius of each circle.
The inner circle A passes through X, so its radius is \(OX = r\).
The middle circle B passes through Y, so its radius is \(OX + XY = r + r = 2r\).
The outer circle C passes through P, so its radius is \(OX + XY + YP = r + r + r = 3r\).

Step 3: Find the area of the ring between the inner and middle circles.
This ring's area is the middle circle's area minus the inner circle's area.
\[ \text{Area}_1 = \pi (2r)^2 - \pi r^2 = \pi(4r^2 - r^2) = 3\pi r^2 \]

Step 4: Find the area of the ring between the middle and outer circles.
This ring's area is the outer circle's area minus the middle circle's area.
\[ \text{Area}_2 = \pi (3r)^2 - \pi (2r)^2 = \pi(9r^2 - 4r^2) = 5\pi r^2 \]

Step 5: Find the required ratio.
\[ \frac{\text{Area}_1}{\text{Area}_2} = \frac{3\pi r^2}{5\pi r^2} = \frac{3}{5} \]
The factor \(\pi r^2\) cancels from the numerator and denominator, so the actual value of \(r\) never matters, only the ratio of the radii does. Options (A), (B) and (D) would come from mismatching which ring is placed on top, or from a slip while squaring the radii.

Final Answer:
The required ratio is 3 : 5. \[ \boxed{\dfrac{3}{5}} \]