Question
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 :
A.
$$\frac{1}{2}\sqrt {c\left( {{g^2} + {f^2} - c} \right)} $$
B.
$$\sqrt {c\left( {{g^2} + {f^2} - c} \right)} $$
C.
$$c\sqrt {{g^2} + {f^2} - c} $$
D.
$$\frac{{\sqrt {{g^2} + {f^2} - c} }}{c}$$
Answer :
$$\sqrt {c\left( {{g^2} + {f^2} - c} \right)} $$
Solution :
Area of quadrilateral $$ = 2$$ [ area of $$\Delta OAC$$ ]
$$\eqalign{ & = 2.\frac{1}{2}OA.AC \cr & = \sqrt {{S_1}} .\sqrt {{g^2} + {f^2} - c} \cr} $$
Point is $$\left( {0,\,0} \right) \Rightarrow {S_1} = c,$$
$$\therefore $$ Area $$ = \sqrt {c\left( {{g^2} + {f^2} - c} \right)} $$
Area of quadrilateral $$ = 2$$ [ area of $$\Delta OAC$$ ]
$$\eqalign{ & = 2.\frac{1}{2}OA.AC \cr & = \sqrt {{S_1}} .\sqrt {{g^2} + {f^2} - c} \cr} $$
Point is $$\left( {0,\,0} \right) \Rightarrow {S_1} = c,$$
$$\therefore $$ Area $$ = \sqrt {c\left( {{g^2} + {f^2} - c} \right)} $$