Find the tangents to the circle $x^2 + y^2 + 2x + 2y = 7$ inclined at $45^\circ$ to the positive direction of the $X$ axis.
Equation of given circle: $\;\;\;$ $x^2 + y^2 + 2x + 2y = 7$
i.e. $\;$ $x^2 + y^2 + 2x + 2y - 7 = 0$ $\;\;\; \cdots \; (1)$
Comparing with the standard equation of circle $\;$ $x^2 + y^2 + 2gx + 2fy + c = 0$ $\;$ gives
$g = 1, \; f = 1, \; c = -7$
Radius of the circle $= r = \sqrt{g^2 + f^2 - c} = \sqrt{1 + 1 + 7} = 3$
Inclination of the required tangent to the positive direction of $X$ axis $= \theta = 45^\circ$
$\therefore \;$ Slope of tangent $= m = \tan 45^\circ = 1$
Let the equation of the required tangent be $y = mx + c_1$
Condition that the line $\;$ $y = mx + c_1$ $\;$ is a tangent to the circle $\;$ $x^2 + y^2 + 2gx + 2fy + c = 0$ $\;$ is
$c_1 = \pm r \sqrt{1 + m^2}$
i.e. $\;$ $c_1 = \pm 3 \sqrt{1 + 1^2} = \pm 3 \sqrt{2}$
$\therefore \;$ The equations of the tangents are
$y = x + 3 \sqrt{2}$ $\;$ and $\;$ $y = x - 3 \sqrt{2}$