Tangent and normal are drawn at P(16, 16) on the parabola y ^ 2 =...

Equation of tangent at $\left({x}_{1}, {y}_{1}\right)$ is $y{y}_{1} = 2a\left(x + {x}_{1}\right)$

Equation of normal at \left({x}_{1}, {y}_{1}\right)$is$y - {y}_{1} = - \dfrac{{y}_{1}}{2a} ( x - {x}_{1} )$<br /><br />Here tangent and normal passes through$P ( 16, 16 )$<br /><br />$∴$Equation of tangent is$2y = x + 16$<br /><br />Equation of normal is$y + 2x = 48$<br /><br />Tangent and normal intersect x-axis at$A\left(- 16, 0\right)$and$B ( 24, 0 )$<br /><br />Since,$APB$is a right angled triangle, hence$AB$is the diameter for the circle.</p><p>Also mid point of$AB$is center of the circle.<br /><br />Hence Centre$= \left(4, 0\right)$<br /><br />Slope of$PC = \dfrac{4}{3}$<br /><br />Slope of$PB$$= - 2$

$tan \theta = \left|\dfrac{\dfrac{4}{3} + 2}{1 - 2 \times \dfrac{4}{3}}\right| = \left|\dfrac{10}{- 5}\right| = 2