An equation of a tangent common to the parabolas $y^{2} = 4 x$ and $x^{2} = 4 y$ is

x - y + 1 = 0

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x + y - 1 = 0

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x + y + 1 = 0

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y = 0

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Solution

The correct option is A $x + y + 1 = 0$
Any tangent to $y^{2} = 4 x$ is $y = m x + \frac{1}{m}$ , tangent to $x^{2} = 4 y$ with slope $m$ is
$x = (\frac{1}{m}) y + m$ or $y = m x - m^{2}$
So $\frac{1}{m} = - m^{2} \Rightarrow m^{3} = - 1 \Rightarrow m = - 1$
and the equation of common tangent is $x + y + 1 = 0$

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The equation of the parabola whose focus is the point (0, 0) and the tangent at the vertex is x-y+1=0 is

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