Multiple choice

PQ is a chord of circle The tangent XR at X on the circle cuts PQ produced at R If XR = 12 cm, PQ = x cm, QR = x - 2 cm then x in cm is

  1. 6

  2. 7

  3. 10

  4. 14
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C Correct answer
Explanation

By the tangent-secant theorem, XR^2 = QR * PR. Here, XR = 12, QR = x-2, PR = PQ + QR = x + (x-2) = 2x-2. So, 12^2 = (x-2)(2x-2). 144 = 2(x-2)(x-1). 72 = x^2 - 3x + 2. x^2 - 3x - 70 = 0. (x-10)(x+7) = 0. Since x must be positive, x = 10.

AI explanation

By the tangent secant theorem, XR^2 = RQ * RP. Substituting the given values gives 12^2 = (x - 2) * ((x - 2) + x), which simplifies to 144 = 2x^2 - 6x + 4. Solving the quadratic equation 2x^2 - 6x - 140 = 0 yields x^2 - 3x - 70 = 0, giving x = 10 or x = -7; since length must be positive, x = 10.

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