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The line among the following which touches the parabola $y^2=4 a x$, is
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Verified Answer
The correct answer is:
$x-m y+a m^2=0$
Given equation of parabola is $y^2=4 a x$
Let the equation of line be $y=m x+c$
If this line touches the parabola, then
$\begin{aligned}
& \qquad c=\frac{a}{m} \\
& \therefore \quad y=m x+\frac{a}{m} \Rightarrow m y=m^2 x+a \\
& \text { replacing } m \text { by } \frac{1}{m} \text {, we get } \\
& \quad m y=x+a m^2 \\
& \Rightarrow \quad x-m y+a m^2=0
\end{aligned}$
Let the equation of line be $y=m x+c$
If this line touches the parabola, then
$\begin{aligned}
& \qquad c=\frac{a}{m} \\
& \therefore \quad y=m x+\frac{a}{m} \Rightarrow m y=m^2 x+a \\
& \text { replacing } m \text { by } \frac{1}{m} \text {, we get } \\
& \quad m y=x+a m^2 \\
& \Rightarrow \quad x-m y+a m^2=0
\end{aligned}$
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