Equation of tangent line of circle $(x-10)^2+(y-10)^2=100$ at point (30,10)

Question

I tried to solve using a formula in my textbook, however it is just appropriate if the point is on the circle. I just realized that the point (30,10) is outside the circle. Please your help.

Answer 1

Let $y=m(x-30)+10$ . We have $$(x-10)^2+(m(x-30))^2=100$$ as a result $$(m^2+1)x^2-(60m+20)x+900m=0$$ and $$\Delta=(60m+20)^2-3600m(m^2+1)=0$$

Answer 2

$y-10=m(x-30)$ is an equation of the tangent line or $mx-y-30m+10=0$.

Thus, $\frac{|10m-10-30m+10|}{\sqrt{m^2+1}}=10$ or $m^2=\frac{1}{3}$ and the rest is smooth.