The area of the quadrilateral formed by the tangents from the point $(4,5)$ to the circle $x^2+y^2-4 x-2 y-11=0$, with a pair of radii joining the points of contact of these tangents is

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Aicharya · Accepted Answer

Equation of given circle $ \begin{aligned} & & x^2+y^2-4 x-2 y-11 & =0 \\ \Rightarrow & & (x-2)^2+(y-1)^2 & =16 \end{aligned} $

From the figure, the area of quadrilateral $P A C B$ $ \begin{aligned} & =2 \times \text { Area of } \triangle P A C \\ & =2 \times \frac{1}{2} \times 4 \times(A P) \end{aligned} $ and $ \begin{aligned} A P & =\text { length of tangent } \\ & =\sqrt{16+25-16-10-11}=2 \end{aligned} $ Therefore area of the required quadrilateral $ =8 \mathrm{sq} \text {. units } $

The area of the quadrilateral formed by the tangents from the point $(4,5)$ to the circle $x^2+y^2-4 x-2…

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