Find the equation of the tangent lines to the circle $x^{2} + y^{2} = 9$ which are parallel to the line $2 x + y - 3 = 0$ .

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Solution

Equation of circle is $x^{2} + y^{2} = 9$
equation of tangent parallel to $2 x + y - 3 = 0$ is $2 x + y + λ = 0$
its distance from centre $=$ radius of the circle.
radius of circle $= 3$ units.
$∴$ $\frac{| 2 \times 0 + 0 + λ |}{\sqrt{4 + 1}} = 3$
$\Rightarrow λ = 3 \sqrt{5}$
$∴$ Equation of tangent is $2 x + y + 3 \sqrt{5} = 0$

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