Question

PQ and RS are common tangents to two circles intersecting at A and B. A and B, when produced on both sides, meet the tangents PQ and RS at X and Y, respectively. If \(AB = 3\) cm and \(XY = 5\) cm, then PQ is:

• A. 4 cm
• B. 2 cm
• C. 3 cm
• D. 6 cm

Hint:

For two circles intersecting at points A and B, the common tangents PQ and RS satisfy the relation \(PQ^2 = XY^2 - AB^2\).

Answer 1

The Correct Option is A

Given two circles intersecting at points A and B. PQ and RS are common tangents touching the circles. Points X and Y are where the lines through A and B meet the tangents PQ and RS respectively.
From the properties of intersecting circles and their common tangents, the length \(PQ\) can be found by: \[ PQ = XY - AB = 5 - 3 = 2 \quad \text{or} \quad PQ = XY + AB = 5 + 3 = 8 \] However, from the figure and properties of tangents, the correct relation is: \[ PQ^2 = XY^2 - AB^2 = 5^2 - 3^2 = 25 - 9 = 16 \implies PQ = 4 \text{ cm} \]