Question

In the circle given below, let OA = 1 unit, OB = 13 unit and PQ \(\perp\) OB. Then, the area of the triangle PQB (in square units) is :

• A. \(24\sqrt{2}\)
• B. \(24\sqrt{3}\)
• C. \(26\sqrt{2}\)
• D. \(26\sqrt{3}\)

Hint:

OB is a diameter, so angle OPB is a right angle; the altitude PA from the right angle satisfies PA^2 = OA x AB.

Answer 1

The Correct Option is B

From the figure the circle passes through O and B with OB as a diameter, so \(\angle OPB=90^\circ\).
PA is the altitude on the hypotenuse OB: \(PA^2=OA\cdot AB=1\times12=12\), so \(PA=2\sqrt3\) and \(PQ=4\sqrt3\).
Height of \(\triangle PQB\) from B to PQ is \(AB=12\).
Area \(=\frac12\times4\sqrt3\times12=24\sqrt3\).
Answer: (2) \(24\sqrt{3}\)