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  • Tangents are drawn to the circle from a point A lying on the x -axis. The tangents cut

Tangents are drawn to the circle x^2+y^2=32 from a point A lying on the x -axis. The tangents cut the y-axis at points B and C, then the coordinate (s) of A such that the area of \triangle A B C is minimum may be

Option: 1

(8 \sqrt{2}, 0)


Option: 2

(8,0)


Option: 3

(-8,0)


Option: 4

(-8 \sqrt{2}, 0)


Answers (1)

best_answer

\begin{aligned} & O A=4 \sqrt{2} \sec \alpha \\ & B C=2 O B=8 \sqrt{2} \operatorname{cosec} \alpha \end{aligned}

\begin{aligned} \text { Area of }\triangle A B C & =\frac{1}{2} \times O A \times B C \\ & =\frac{1}{2} \times 4 \sqrt{2} \sec \alpha \times 8 \sqrt{2} \operatorname{cosec} \alpha \\ & =\frac{64}{\sin 2 \alpha} \end{aligned}

Which is minimum when \sin 2 \alpha is maximum ie, when \alpha=\frac{\pi}{4}

\begin{aligned} & \therefore O A=4 \sqrt{2} \times \sqrt{2}=8 \\ & \therefore \quad A \equiv(8,0) \text { \text { and symmetrically on the other side } } \\ & \: \: \: \: \: \: \quad A^{\prime} \equiv(-8,0) \end{aligned}

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Riya

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