  Question

The range of value of $$P$$ such that the angle $$\theta$$ between the pair of tangents drawn from the point $$(p.0)$$ to the circle $${ x }^{ 2 }+{ y }^{ 2 }=1$$ lies in $$\displaystyle \left( \frac { \pi }{ 3 } ,\pi \right)$$, is

A
(2,1)(1,2)  B
(3,2)(2,3)  C
(0,2)  D
None of these  Solution

The correct option is C $$\left( -2,-1 \right) \cup \left( 1,2 \right)$$We have $$\displaystyle \frac { \pi }{ 3 } <\theta <\pi$$$$\Rightarrow\displaystyle \frac { \pi }{ 6 } <\frac { \theta }{ 2 } <\frac { \pi }{ 2 } \Rightarrow \frac { 1 }{ 2 } <\sin { \left( \frac { \theta }{ 2 } \right) } <1$$$$\displaystyle\Rightarrow \frac { 1 }{ 2 } <\frac { 1 }{ a } <1\quad \quad \left[ \because \sin { \left( \frac { \theta }{ 2 } \right) } =\frac { 1 }{ a } \right]$$$$\therefore 1<a<2$$There can be symmetrical points on the negative x-axis too. Hence, we have $$a\in \left( -2,-1 \right) \cup \left( 1,2 \right)$$ Mathematics

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