question_answer

Tangents are draw to the circle \[{{x}^{2}}+{{y}^{2}}=50\] from a point \['p'\] lying on the x-axis. These tangents meet the y-axis at point \['{{P}_{1}}'\]and \['{{P}_{2}}'\]possible coordinates of \['P'\]so that area of triangle \[P{{P}_{1}}{{P}_{2}}\] is minimum, are

A) \[\left( 10,0 \right)\]

B) \[\left( 10\sqrt{2,0} \right)\]

C) \[\left( -10\sqrt{2,0} \right)\]

D) none of these

Correct Answer: A

Solution :

\[OP=5\sqrt{2}\sec \theta ,O{{P}_{1}}=5\sqrt{2}\cos ec\theta \]

\[\Delta P{{P}_{_{1}}}{{P}_{2}}=\frac{100}{\sin 2\theta },{{\left( \Delta P{{P}_{1}}{{P}_{2}} \right)}_{\min }}=100\]

\[\Rightarrow \theta =\pi /4\Rightarrow OP=10\]

\[\Rightarrow P=\left( 10,0 \right),\left( -10,0 \right)\]