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Position of a Line W.R.T Hyperbola

lf the tangen...

Question

lf the tangent at $P$ on the circle $x^{2} + y^{2} = a^{2}$ cuts two parallel tangents of the circle at $A$ and $B$ then $P A . P B =$

a

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a^{2}

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2 a

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2 a^{2}

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Solution

The correct option is C $a^{2}$
Let p $(a c o s θ, a s i n θ)$ is a point on $x^{2} + y^{2} = a^{2}$ on circle $x^{2} + y^{2} = a^{2}$
and Let x=a and x=-a are two porated targets of $x^{2} + y^{2} = a^{2}$ circle
so $e q^{n}$ of tangent at p ca casa, business is given by
x cos a+y sin a =a
and x=a, and x=-a , are two parallel tangent
$∴$ from 1& 2
$y = \frac{a (1 - c o s a)}{s i n a}$ , x=a $\Rightarrow$ $A (a, a t a n^{a} /_{2})$
from 1,3,
s=-a and $y = \frac{a (1 + c a s a)}{s i n a} \Rightarrow B (- a, a c o t^{a} /_{2}) .$
$a t a n^{a} /_{2} = P A = \sqrt{a^{2} (c o s a - 1)^{2} + a^{2} (s i n a - t a r^{a} /_{2})^{2}}$
$a c o t^{a} /_{2} = P B = \sqrt{a^{2} (c o s a + 1)^{2} + a^{2} (s i n a - c o t^{a} /_{2})^{2}}$
so , $P A, P B = a^{2}$

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lf the tangent at P on the circle x2+y2=a2 cuts two parallel tangents of the circle at A and B then PA. PB=

lf the tangent at $P$ on the circle $x^{2} + y^{2} = a^{2}$ cuts two parallel tangents of the circle at $A$ and $B$ then $P A . P B =$