A common tangent is drawn to the circle $x^{2} + y^{2} = c^{2}$ and the parabola $y^{2} = 4ax$. If t

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A common tangent is drawn to the circle $x^{2} + y^{2} = c^{2}$ and the parabola $y^{2} = 4ax$. If the angle which this tangent makes with the axis of $x$ is $\pi/4$ then the relationship between $a$ and $c$ ( $a,c > 0$ ) is:

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Sol. [Key Concept: The slope-1 tangent to $y^{2} = 4ax$ is $y = x + a$; impose tangency to the circle $x^{2} + y^{2} = c^{2}$ to relate $a$ and $c$.]Equation of tangent is$$yt = x + at^{2} $$slope $= \frac{1}{t} = tan\frac{\pi}{4} = 1 \Rightarrow t = 1$$$\begin{matrix} & \ \Rightarrow \ y = x + a \ & \ \Rightarrow \ c = \frac{|0 + a - 0|}{\sqrt{2}} = \frac{a}{\sqrt{2}} \Rightarrow a = \sqrt{2}c \end{matrix}$$

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