Multiple choice

A circle C1 passes through the origin O and has diameter 4 on the positive x-axis. The line y = 2x gives a chord OA of a circle C1. Let C2 be the circle with OA as a diameter. If the tangent to C2 at the point A meets the x-axis at P and y-axis at Q, then QA : AP is equal to:

  1. 1 : 4

  2. 1 : 5

  3. 2 : 5

  4. 1 : 3
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A Correct answer
Explanation

C1 has diameter 4 on x-axis, center (2,0), radius 2. Equation: (x-2)^2 + y^2 = 4. Line y=2x intersects C1 at A. Substituting: (x-2)^2 + 4x^2 = 4 -> x^2 - 4x + 4 + 4x^2 = 4 -> 5x^2 - 4x = 0. A is (4/5, 8/5). C2 has OA as diameter, center (2/5, 4/5), radius = sqrt((2/5)^2 + (4/5)^2) = sqrt(20/25) = sqrt(4/5). Tangent at A to C2 leads to the ratio 1:4.

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