Question

PQ and RS are common tangents to two circles intersecting at points 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:

Use the relationship between lengths of common tangents and chord segments in intersecting circles.

Answer 1

The Correct Option is A

Given two circles intersect at points \(A\) and \(B\). The common tangents \(PQ\) and \(RS\) are such that when lines through \(A\) and \(B\) meet tangents at points \(X\) and \(Y\), the segment \(AB = 3 \text{ cm}\) and \(XY = 5 \text{ cm}\).
By the property of intersecting circles and their common tangents, the length \(PQ\) can be found using: \[ PQ = \frac{XY^2 - AB^2}{2 \times AB} \] However, since the problem is classic, the relation simplifies to: \[ PQ = \sqrt{XY^2 - AB^2} \] Calculate: \[ PQ = \sqrt{5^2 - 3^2} = \sqrt{25 - 9} = \sqrt{16} = 4 \text{ cm} \]