Equation of Tangent to Circle Centered at Origin/Proof 1

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Theorem

Let $\CC$ be a circle whose radius is $r$ and whose center is at the origin of a Cartesian plane.

Let $P = \tuple {x_1, y_1}$ be a point on $\CC$.

Let $\TT$ be a tangent to $\CC$ passing through $P$.


Then $\TT$ can be defined by the equation:

$x x_1 + y y_1 = r^2$


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 $\tuple {0, 0}$:

\(\ds y - y_1\) \(=\) \(\ds -\dfrac {x_1} {y_1} \paren {x - x_1}\)
\(\ds \leadsto \ \ \) \(\ds y_1 \paren {y - y_1}\) \(=\) \(\ds -x_1 \paren {x - x_1}\)
\(\ds \leadsto \ \ \) \(\ds x x_1 + y y_1\) \(=\) \(\ds x_1^2 + y_1^2\)
\(\ds \) \(=\) \(\ds r^2\) as $\tuple {x_1, y_1}$ is on $\CC$

$\blacksquare$