Question

Find the condition for the line $y=mx+c$ to be a tangent to the parabola $x^2=4ay$.

Answer 1

$y=mx+c$

$x^2=4ay$

for line $y=mx+c$ to be tangent to $x^2=4ay$

slope of $y=mx+c$ should be equal to

slope of $x^2=4ay$ at point of tang ency

Let point of tangency $=(x_1,y_1)$

${\begin{array}{c}\frac{dy}{dx}\end{array}|}_{(x_1,y_1)}$ of $x^2=4ay$ should be equal to $m$

$\Rightarrow \frac{2x_1}{4a}=m$

$\Rightarrow x_1=2am$

putting $x_1=2am$ in $x^2=4ay$ we get, $y_1=a{m}^2$

putting $(x_1,y_1)$ in line $y=mx+c$ we get $a{m}^2=2a{m}^2+c$

$c=-a{m}^2$