Multiple choice

If $ OA$ and $OB$ are the tangents from the origin to the circle $x^{2} + y^{2} + 2gx + 2fy + c = 0,$ and $C$ is the centre of the circle, the area of the quadrilateral $OACB$ is

  1. $\displaystyle \frac{1}{2} \sqrt{c(g^{2}+f^{2}-c}$
  2. $\sqrt {c (g^{2} + f^{2} -c)}$
  3. $ c \sqrt { (g^{2} + f^{2} -c)}$
  4. $\displaystyle \frac{\sqrt{g^{2}+f^{2}-c}}{c}$
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B Correct answer
Explanation

The centre is C(-g, -f), and the radius is sqrt(g^2 + f^2 - c). The tangent length from O is sqrt(c). The quadrilateral consists of two right triangles, so its area is radius times tangent length, giving sqrt(c(g^2 + f^2 - c)).

AI explanation

The centre of the circle is (-g, -f) and its radius is sqrt(g^2 + f^2 - c). The length of tangent OA is sqrt(CO^2 - r^2) = sqrt(g^2 + f^2 - (g^2 + f^2 - c)) = sqrt(c). The area of quadrilateral OACB is twice the area of right triangle OAC, calculated as 2 * (1/2) * OA * CA = sqrt(c) * sqrt(g^2 + f^2 - c) = sqrt(c(g^2 + f^2 - c)).

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