Find the equation of tangent to the circle $x^{2} + y^{2} = 4$ and parallel to the line x+2y+3=0.

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Solution

The tangents to the circle $x^{2} + y^{2} = 4$ in the slope is given by $y = m x \pm 2 \sqrt{1 + m^{2}}$
These will be parallel to the line $x + 2 y + 3 = 0$
if $m = \frac{- 1}{2} =$ slope of the line
Required tangents are $y = \frac{- 1}{2} x \pm 2 \sqrt{1 + \frac{1}{4}}$
$2 y + x = \pm 2 \sqrt{5}$

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