At the point of intersection of the rectangul | vedclass

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Question

Let $ (x_1, y_1)$  be the point of intersection $ \Rightarrow $ $y_1^2\, = \,4a{x_1}\,$ and $ x_1y_1 = c^2$
$Y^2 = 4ax $      &nb

Vedclass · Accepted Answer

$	heta = tan^{-1} (- 2 tan\phi )$

Vedclass · Answer

Let $ (x_1, y_1)$  be the point of intersection $ \Rightarrow $ $y_1^2\, = \,4a{x_1}\,$ and $ x_1y_1 = c^2$
$Y^2 = 4ax $                     $xy = c^2$
  $\frac{{dy}}{{dx}}\, = \,\frac{{2a}}{y}\,$                  $\frac{{dy}}{{dx}}\, = \, - \,\,\frac{y}{x}\,\,$ 
${\frac{{dy}}{{dx}}_{({x_1},{y_1})}}\, = \,\,\,	an \phi \,\,\, = \,\,\,\frac{{2a}}{{{y_1}}}\,$                  ${\frac{{dy}}{{dx}}_{({x_1},{y_1})}}\, = \,	an 	heta \, = \, - \,\frac{{{y_1}}}{{{x_1}}}$
$\frac{{	an 	heta }}{{	an \phi }}\,\, = \,\,\frac{{ - {y_1}/{x_1}}}{{2a/{y_1}}}\,\, = \,\,\frac{{ - y_1^2}}{{2a{x_1}}}\,\, = \,\, - \,\,\frac{{4a{x_1}}}{{2a{x_1}}}\,\, = \,\, - 2$
$\Rightarrow$ $	heta = 	an^{-1} (- 2 	an\phi )$

At the point of intersection of the rectangular hyperbola $ xy = c^2 $ and the parabola $y^2 = 4ax$ tangents to the rectangular hyperbola and the parabola make an angle $ \theta $ and $ \phi $ respectively with the axis of $X$, then

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