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If the area of the quadrilateral formed by the tangents from the origin to the circle x^2+y^2+6 x-10 y+c=0 and the pair of radii at the points of contact of these tangents to the circle is 8 \: \mathrm{sq} unit, then the value of c is

Option: 1

2


Option: 2

4


Option: 3

16


Option: 4

32


Answers (1)

best_answer

Let OA, OB be the tangents from the origin to the given circle with centre C(-3,5)$ and radius $\sqrt{(9+25-c)}

The area of the quadrilateral OACB 

                                 \begin{aligned} & =2 \times \text { area of } \triangle O A C \\ \\& =2 \times \frac{1}{2} \times O A \times A C \end{aligned}

Now OA = length of tangent from the origin to the given circle =\sqrt{c} \text { and } A C= radius of the circle = \sqrt{34-c}

\begin{array}{lc} \text { So } & \sqrt{c} \sqrt{34-c}=8 \text { (given) } \\ \\\Rightarrow & c(34-c)=64 \\ \\\Rightarrow & c^2-34 c+64=0 \\ \\\Rightarrow & c=2 \text { or } c=32 \end{array}

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Ritika Jonwal

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