A straight line intersects two concentric circles (circles having the same centre) and forms chords of lengths 4 cm and 16 cm on the smaller and larger circles respectively. What is the difference, in square centimetres, between the squares of the radii of the larger and smaller circles?

Introduction / Context:
This question examines your understanding of the relationship between a chord length, the radius of a circle and the perpendicular distance from the centre to the chord. When the same line cuts two concentric circles, the perpendicular distance from the common centre to that line is the same for both chords, which allows us to relate the two radii through a simple algebraic relation.

Given Data / Assumptions:

• Two circles are concentric, so they share the same centre.

• A straight line cuts the smaller circle forming a chord of length 4 cm.

• The same line cuts the larger circle forming a chord of length 16 cm.

• Let r be the radius of the smaller circle and R be the radius of the larger circle.

• Let d be the perpendicular distance from the common centre to the line.

Concept / Approach:
For any circle, if a chord of length L is at a perpendicular distance d from the centre and the radius is s, then by right triangle geometry we have (L / 2)^2 + d^2 = s^2. We apply this relation separately to the smaller and larger circles using their respective chord lengths and radii but the same distance d. Subtracting the two equations will eliminate d^2, leaving us with an expression for R^2 − r^2, which is exactly what we need.

Step-by-Step Solution:
For the smaller circle, chord length = 4 cm, radius = r, distance = d. Using the relation: (4 / 2)^2 + d^2 = r^2. So, 2^2 + d^2 = r^2, which gives 4 + d^2 = r^2. For the larger circle, chord length = 16 cm, radius = R, same distance d. Apply the relation: (16 / 2)^2 + d^2 = R^2. So, 8^2 + d^2 = R^2, which gives 64 + d^2 = R^2. We want R^2 − r^2. Subtract the first equation from the second: (64 + d^2) − (4 + d^2) = R^2 − r^2. This simplifies to 60 = R^2 − r^2. Therefore, the difference between the squares of the radii is 60 square centimetres.

Verification / Alternative check:
To check, we can imagine a particular numerical example that satisfies these equations. Suppose we let d^2 = 4, then from 4 + d^2 = r^2 we get r^2 = 8. From 64 + d^2 = R^2 we then get R^2 = 68. The difference R^2 − r^2 is 68 − 8 = 60, which matches our general result and shows that the method is consistent and independent of the actual value of d, as long as the geometry is correctly applied.

Why Other Options Are Wrong:
The values 120, 90 and 240 square centimetres do not follow from the derived equations. They would result from errors such as using the full chord length instead of half the length in the Pythagoras relation, or incorrectly subtracting the equations. Only 60 is consistent with the relationships for both chords and radii with a common perpendicular distance from the centre.

Common Pitfalls:
Common mistakes include forgetting to halve the chord length when applying the right triangle relation, mixing up r and R, or assuming that the distances from the centre to each chord are different, which they are not, because the same line cuts both circles. Another pitfall is to try to assign numerical values to r and R directly without using the elimination method, which can complicate the algebra unnecessarily.

Final Answer:
The difference between the squares of the radii of the two concentric circles is 60 square centimetres.