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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)
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B
(3,2)(2,3)
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C
(0,2)
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D
None of these
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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 $$a\in \left( -2,-1 \right) \cup \left( 1,2 \right) $$ 

389345_190721_ans_a27d96e3c03b4ede9f2a1f7389a216ce.png

Mathematics

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