Find $k$ , if the line $x - y + 3 = 0$ touches the parabola $y^{2} = 4 k x$

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Solution

The given line is
$x - y + 3 = 0$
$\Rightarrow$ $y = x + 3$ ....(i)
Comparing with $y = m x + c$ , we get
$m = 1, c = 3$
The given parabola is
$y^{2} = 4 k x$ .....(ii)
If line (i) touches the parabola (ii), then
$c = \frac{k}{m}$
$\Rightarrow$ $3 = \frac{k}{1}$
$\Rightarrow k = 3$

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