Find the equation of the tangent to the parabola $y^{2} = 4 a x$ when slope is $2$

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Solution

$y^{2} = 4 a x . . . . (1)$
$2 y y^{'} = 4 a$
$\Rightarrow$ $y^{'} = \frac{2 a}{y}$ (slope tangent $= 2$ )
$\Rightarrow$ $\frac{2 a}{y} = 2$
$\Rightarrow$ $y = a$
put $y = a$ in (1)
$a^{2} = 4 a x \Rightarrow x = \frac{a}{4}$
Tangent $y - a = 2 (x - \frac{a}{4})$
$y = 2 x + \frac{a}{2}$

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