A line cuts two concentric circles. The length of chords formed by this line on the circles is 6 cm and 18 cm. Find the difference in the squares of the radii of two circles.

Correct — A. One line, two chords, and the same perpendicular distance d from the shared centre.

The perpendicular from the centre bisects a chord, so the half-chords are 18 ⁄ 2 = 9 cm and 6 ⁄ 2 = 3 cm.

Larger circle: R² = d² + 9² = d² + 81
Smaller circle: r² = d² + 3² = d² + 9

R² − r² = 81 − 9 = 72 → option (a). The distance d cancels, which is why the question can ask this without giving either radius.

Why the others are wrong

• (b)
  90 — 90 is 81 + 9, the two squared half-chords added rather than subtracted. That sum equals R² + r² − 2d², which the question does not ask for.
• (c)
  120 — The half-chords are fixed at 3 cm and 9 cm, so the difference of squares is 81 − 9 and nothing else. Reaching 120 would need half-chords no halving of 6 and 18 can give.
• (d)
  60 — 60 does not come out of 9² − 3². A common route to a too-small figure is halving only the longer chord, but even 9² − 6² is 45.

The tool is the theorem that a perpendicular from the centre bisects the chord, and its consequence: a right triangle whose legs are d and the half-chord and whose hypotenuse is the radius.

For concentric circles the centre is shared, so a single line cutting both stands at one distance d from that centre. Write the theorem twice and d carries the same value in both equations.

Subtracting removes it: R² − r² = (d² + 81) − (d² + 9) = 72. The question asks for the difference of squares precisely because neither radius is determined on its own.

Key facts

• A perpendicular dropped from the centre of a circle to a chord bisects that chord.
• Radius² = (distance from centre to the chord)² + (half-chord)².
• Chords of 18 cm and 6 cm cut by one line on concentric circles force R² − r² = 9² − 3² = 72.

Common traps

• Using the full chords 18 and 6 instead of the half-chords 9 and 3, which gives 288.
• Trying to pin down R and r individually, which the data will not support.
• Assuming the line passes through the centre, which would make both chords diameters.

No directly related past PYQ was found.