Tangent to a circle equation

Question

So I wanted to make a program in desmos which shows a tangent to a circle of user defined radius r at the point q (y co ordinate). I've achieved this so far but of course due to negative square roots, etc. I can only have the tangent line be shown in the first quadrant. It dissapears in quadrant two and three, and doesn't work in quadrant four. Can someone please make corrections.

Answer 1

Let the circle be given by

$ x^2 + y^2 = r^2 $

A point $(p, q)$ on the circle is parameterized by the angle $\theta$ as follows

$ (p, q) = r (\cos(\theta), \sin(\theta) ) $

The tangent at $(p, q)$ has a unit normal vector equal to $(\cos(\theta),\sin(\theta) )$

Therefore, the equation you want to plot is

$ \cos(\theta) ( x - p) + \sin(\theta)( y - q ) = 0 $

This is shown below for $r = 2.7$ and $\theta = \dfrac{\pi}{3} $