What is the common tangent to the parabola y^ | vedclass

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(D) The equation of any tangent to the parabola $y^{2} = 8ax$ is $y = mx + \frac{2a}{m}$.For this line to be a tangent to the circle $x^{2} + y^{2} =

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$y = x + 2a$

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(D) The equation of any tangent to the parabola $y^{2} = 8ax$ is $y = mx + \frac{2a}{m}$.For this line to be a tangent to the circle $x^{2} + y^{2} = 2a^{2}$,the perpendicular distance from the center $(0, 0)$ to the line $mx - y + \frac{2a}{m} = 0$ must be equal to the radius $r = a\sqrt{2}$.Using the distance formula: $\frac{|\frac{2a}{m}|}{\sqrt{m^{2} + 1}} = a\sqrt{2}$.Squaring both sides: $\frac{4a^{2}}{m^{2}(m^{2} + 1)} = 2a^{2}$.$\frac{2}{m^{2}(m^{2} + 1)} = 1 \Rightarrow m^{4} + m^{2} - 2 = 0$.$(m^{2} + 2)(m^{2} - 1) = 0$. Since $m$ must be real,$m^{2} = 1$,so $m = \pm 1$.Substituting $m = 1$ into the tangent equation: $y = x + \frac{2a}{1} = x + 2a$.Thus,$y = x + 2a$ is a common tangent.

What is the common tangent to the parabola $y^{2} = 8ax$ and the circle $x^{2} + y^{2} = 2a^{2}$?

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