Let $x^{2} + y^{2} - 4 x - 2 y - 11 = 0$ be a circle. A pair of tangents from the point (4, 5) with a pair of radii form a quadrilateral of area ___.

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Solution

The equation of the circle is $x^{2} + y^{2} - 4 x - 2 y - 11 = 0$ .
Its center is (2, 1) and radius is $\sqrt{4 + 1 + 11} = 4 = B C$ .

$A C = 2 \sqrt{5} . . . (F r o m D i s t a n c e f o r m u l a) ∴ A B = 2 . . . (F r o m r i g h t Δ A B C) ∴ A r e a o f q u a d r i l a t e r a l A B C D = 2 \times (A r e a o f Δ A B C) = 2 \times \frac{1}{2} \times A B \times B C = 2 \times \frac{1}{2} \times 2 \times 4 = 8 s q . u n i t s$

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