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

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Solution

If $y = m x + c$ is a tangent
$x^{2} = 4 a y \Rightarrow x^{2} = 4 a (m x + c) \Rightarrow x^{2} - 4 a m x - 4 a c = 0 D = 0 b^{2} - 4 a c = 0 \Rightarrow 16 a^{2} m^{2} - 4 \times - 4 a c = 0 \Rightarrow 16 a^{2} m^{2} + 16 a c = 0 \Rightarrow a^{2} m^{2} + a c = 0$

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