Q.The equations of the tangents to the circle $x^2+y^2=4$ which are parallel to $x+2y+3=0$ are (A) $x-2y=2$ (B) $x+2y=\pm2\sqrt3$ (C) $x+2y=\pm2\sqrt5$ (D) $x-2y=\pm2\sqrt5$

For a line of unknown position, written generally as $y=mx+c$, tangency to the circle $x^2+y^2=a^2$ is decided by comparing it against the tangent-equation pattern at an as-yet-unknown point of contact and eliminating that point's coordinates; the result is a clean condition purely in terms of $m$, $c$, and $a$:

Equivalently, $c=\pm\sqrt{a^2m^2+a^2}$ — meaning that for any slope $m$, there are always exactly two tangent lines to a given circle with that slope, one on each side of it. This condition is the standard tool whenever a tangent's slope is known (or constrained — parallel to a given line, perpendicular to one, or a stated numeric slope) but its exact position is not: substitute the slope, solve for $c$, and the tangent line(s) follow directly. An equivalent and often faster check for the same question is purely …