Equation of Tangent to Circle Centered at Origin/Proof 2

Proof
From the slope-intercept form of a line, the equation of a line passing through $P$ is:
 * $y - y_1 = \mu \paren {x - x_1}$

If this line passes through another point $\tuple {x_2, y_2}$ on $\CC$, the slope of the line is given by:
 * $\mu = \dfrac {y_2 - y_1} {x_2 - x_1}$

Because $P$ and $Q$ both lie on $\CC$, we have:

As $Q$ approaches $P$, we have that $y_2 \to y_1$ and $x_2 \to x_1$.

The limit of the slope is therefore:


 * $-\dfrac {2 x_1} {2 y_1} = -\dfrac {x_1} {y_1}$

The equation of the tangent $\TT$ to $\CC$ passing through $\tuple {x_1, y_1}$ is therefore: