Multiple choice

From a point P(16, 7), tangent PQ and PR are drawn to the circle ${ x }^{ 2 }+{ y }^{ 2 }-2x-4y-20=0$. If C be the center then area of the quadrilateral PQCR will be-

  1. 75

  2. 150

  3. 15

  4. None of these
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A Correct answer
Explanation

Circle: x^2 + y^2 - 2x - 4y - 20 = 0. Center C = (1, 2). Radius r = sqrt(1^2 + 2^2 - (-20)) = sqrt(25) = 5. Point P = (16, 7). Distance CP = sqrt((16-1)^2 + (7-2)^2) = sqrt(15^2 + 5^2) = sqrt(225 + 25) = sqrt(250). Area of triangle PQC = 1/2 * base * height = 1/2 * r * sqrt(CP^2 - r^2) = 1/2 * 5 * sqrt(250 - 25) = 1/2 * 5 * sqrt(225) = 1/2 * 5 * 15 = 37.5. Quadrilateral PQCR consists of two such triangles, so area = 2 * 37.5 = 75.

AI explanation

The center of the circle is C(1, 2) and its radius is the square root of 1 squared plus 2 squared minus negative 20, which equals 5. The length of the tangent from P(16, 7) is found using the formula for the tangent length, giving the square root of 15 squared plus 5 squared minus 5 squared, which equals 15. The quadrilateral consists of two identical right triangles, so the total area is 2 times one half times the radius times the tangent length, yielding 2 times one half times 5 times 15, which is 75.

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