Find the equation of the tangent lines to the circle $x^2 + y^2 = 9$ which are parallel to $2x + y - 3 = 0$.
Equation of given line: $\;\;\;$ $2x + y - 3 = 0$
i.e. $y = -2x + 3$ $\;\;\; \cdots \; (1)$
$\therefore$ $\;$ Slope of given line $= m = -2$
Since the required tangents to the circle are parallel to the given line,
slope of required tangents $= m = -2$ $\;\;\; \cdots \; (2)$
Equation of given circle: $\;\;\;$ $x^2 + y^2 = 9$ $\;\;\; \cdots \; (3)$
Comparing with the standard equation of circle: $\;\;\;$ $x^2 + y^2 = a^2$
we have $\;$ $a^2 = 9$ $\implies$ $a = \pm 3$
Equation of tangent (with slope m) to a circle $\;$ $x^2 + y^2 = a^2$ $\;$ is $\;$ $y = mx \pm a \sqrt{1 + m^2}$
$\therefore$ $\;$ Required equations of tangent lines are
$y = - 2x \pm 3 \sqrt{1 + \left(-2\right)^2}$
i.e. $y = -2x \pm 3 \sqrt{5}$
i.e. $2x + y = \pm 3 \sqrt{5}$