Find pair of tangents from $(3,4)$ to circle $x^{2}+y^{2}=9$
We have to find tangents to circle. I have supposed a line $$y = mx + c.$$ By condition of tangency I have $$c = 3\sqrt{1+m^{2}}.$$ Now as this line passes through $(3,4)$ I have
$$4 = 3m + 3\sqrt{1+m^{2}}.$$
$$ \therefore\quad 4-3m = 3\sqrt{1+m^{2}}$$
Squaring on both sides $$\therefore\quad 16 +9m^{2} -24m = 9 + 9m^{2}$$
$$\therefore\quad 7 -24m = 0$$
now I am stuck here as $(3,4)$ is outside the Circle , there must be two tangents from $(3,4)$ but here $m$ only has one value. Now what to do?
One tangent at x=3 is parallel to y-axis at distance3, with a distance from given point to circle center as 5. Since it has $ m = \infty$ it will not show up in your calculation which assumed $m$ to be non-infinite. I.e., $ m,c $ both go to $\infty$. Other tangent slope $\frac{7}{24}$ ok.