Find the equation of the straight line which touches both the circle $x^{2} + y^{2} = 2 a^{2}$ and the parabola $y^{2} = 8 a x$ .

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Solution

Given, $x^{2} + y^{2} = 2 a^{2}$
& $y^{2} = 8 a x$
equation of tangent to parabola in slope form :
$y = m x + \frac{2 a}{m}$ _- (1)
equation of tangent to circle :
$y = m x + \sqrt{2 a} \sqrt{1 + m^{2}}$ __ (2)
(1) & (2) must be same
$∴ \frac{2 a}{m} = \sqrt{2 a} \sqrt{1 + m^{2}}$
$= m^{4} + m^{2} - 2 = 0$
$∴ m = \pm 1$
$∴$ equation of common tangent
y=x+2a, y=-x-2a

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2

x^{2} + y^{2} = 2 a^{2}

y^{2} = 8 a x

x^{2} + y^{2} = 2 a^{2}

y^{2} = 8 a x

x^{2} + y^{2} = 2 a^{2}

y^{2} = 8 a x

x^{2} + y^{2} = 2 a^{2}

y^{2} = 8 a x

y = \pm (x + 2 a)

The equation of the common tangent to $x^{2} + y^{2} = 2 a^{2}$ and $y^{2} = 8 a x$ is

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