PQ and RS are common tangents to two circles intersecting at A and B. A and B, when produced on both the sides, meet the tangents PQ and RS at X and Y, respectively. If AB = 3 cm and XY = 5 cm, then PQ is ________.
Show answer & explanation
Correct answer: D
Explanation
The correct answer is option 4 i.e. 4 cm.
XY = 5 cm and AB = 3 cm
Here, XA = YB
Also, XY = XA + AB + BY
⇒ 5 = 2XA + 3
⇒ 2XA = 5 - 3 = 2
⇒ XA = 1 cm
Now, XB = XA + AB = 1 + 3 = 4 cm
If a secant and a tangent are drawn to a circle from a common exterior point, then the product of the length of the whole secant segment and its external secant segment is equal to the square of the length of the tangent segment.
PX2 = XA × XB
⇒ PX2 = 1 × 4 = 4
⇒ PX = √4 = 2 cm
Similarly, XQ = 2 cm
PQ = PX + XQ = 2 + 2 = 4 cm
Therefore, the measure of PQ is 4 cm.