Multiple choice

The area of the quadrilaterial formed by the two tangents from $(4,5)$ to the circle ${x}^{2}+{y}^{2} -4x- 2y-11=0$ and a pair of radii is

  1. $2$
  2. $4$
  3. $8$
  4. $16$
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C Correct answer
Explanation

The circle is (x-2)^2 + (y-1)^2 = 11 + 4 + 1 = 16. Radius r = 4. Distance from (4,5) to center (2,1) is sqrt((4-2)^2 + (5-1)^2) = sqrt(4+16) = sqrt(20). Length of tangent L = sqrt(d^2 - r^2) = sqrt(20 - 16) = 2. The area of the quadrilateral formed by two tangents and two radii is 2 * (1/2 * L * r) = L * r = 2 * 4 = 8.

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