Multiple choice

P is a point outside a circle and is 13 cm away from its centre. A secant drawn from the point P intersects the circle at points A and B in such a way that PA = 9 cm and AB = 7 cm. The radius of the circle is

  1. 5 cm

  2. 4 cm

  3. 4.5 cm

  4. 5.5 cm
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A Correct answer
Explanation

Let O be the center, P the external point. OP = 13. Secant PAB, PA = 9, AB = 7, so PB = 16. Power of point P = PA * PB = 9 * 16 = 144. Also, Power = d^2 - r^2, where d=13. 144 = 13^2 - r^2 = 169 - r^2. r^2 = 169 - 144 = 25. r = 5.

AI explanation

The total length of the secant from P to the far side of the circle is PA plus AB, making PB equal to 9 plus 7, which is 16 cm. By the Power of a Point theorem, PA times PB equals the square of the length of the tangent, which also equals the square of the distance to the center minus the square of the radius. Setting up the equation, 9 times 16 equals 13 squared minus r squared, so 144 equals 169 minus r squared. Solving for r gives r squared equals 25, making the radius 5 cm.

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