When two tangents, PA and PB, are drawn to a circle from a point P outside the circle, and the angle between them is 90°, what is the distance from P to the center O of the circle, if the radius is 5 cm?
Show answer & explanation
Correct answer: B
Explanation
The correct answer is option 2 i.e. cm.
Radius = 5 cm, ∠APB = 90°
OA ⟂ PA, OB ⟂ PB, PA = PB = 5 cm
In quadrilateral OAPB: ∠OAP = ∠OBP = 90°, ∠APB = 90° ⇒ ∠AOB = 90°
In right-angle ΔAOB: AB2 = 52 + 52 = 25 + 25 = 50
⇒ AB = √50 = 5√2 cm
In right-angle ΔAPB: AB2 = PA2 + PB2 = 2PA2
⇒ 50 = 2PA2
⇒ PA2 = 25
⇒ PA = 5 cm
In rright-angle ΔOAP: OP2 = OA2 + PA2 = 52 + 52 = 25 + 25 = 50
⇒ OP = √50 = 5√2 cm