The tangent and normal at the point $P\left( a{ t }^{ 2 },2at \right) $ to the parabola ${ y }^{ 2 }=4ax$ meet the x-axis in $T$ and $G$ respectively, then angle at which the tangent at $P$ to the parabola is inclined to the tangent at $P$ to the circle through $P,T,G$ is
-
$\tan ^{ -1 }{ \left( { t }^{ 2 } \right) } $
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$\cot ^{ -1 }{ \left( { t }^{ 2 } \right) } $
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$\tan ^{ -1 }{ \left( { t } \right) } $
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$\cot ^{ -1 }{ \left( { t } \right) } $
C
Correct answer
Explanation
The tangent at P(at^2, 2at) has slope 1/t. The normal has slope -t. The angle between the tangent and the normal is 90 degrees. The circle through P, T, G has the normal as a diameter. The angle between the tangent at P to the parabola and the tangent at P to the circle is tan^-1(t).