Equation of Tangent to Circle Centered at Origin/Proof 1

Proof
From Equation of Straight Line Tangent to Circle we have that for a general circle of radius $r$ and center $\tuple {a, b}$:


 * $y - y_1 = \dfrac {a - x_1} {y_1 - b} \paren {x - x_1}$

is the equation of a tangent $\TT$ to $\CC$ passing through $\tuple {x_1, y_1}$.

Setting the center to $\tuple {0, 0}$, the result follows:


 * $y - y_1 = -\dfrac {x_1} {y_1} \paren {x - x_1}$