Question

Two concentric circles are of radii a and b (a > b) are given. Find the length of the chord of the larger circle which touches the smaller circle.

Answer 1

Given: Two concentric circles with common centre O and radii a (larger) and b (smaller), where a > b. Let AB be a chord of the larger circle that is tangent to the smaller circle at C.

Step-wise calculation:

1. Join O to A and C. Then OA = a and OC = b.

2. Because AB is tangent to the smaller circle at C, OC ⟂ AB; hence OC is perpendicular to AB and therefore bisects AB at C.

3. In right triangle AOC apply Pythagoras:

OA² = OC² + AC² 

⇒ a² = b² + AC²

⇒ `AC = sqrt(a^2 - b^2)`

4. Since C is the midpoint of AB.

AB = 2 × AC 

= `2 xx sqrt(a^2 − b^2)`

The length of the chord AB of the larger circle that touches the smaller circle is `AB = 2sqrt(a^2 - b^2)`.