User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

= Tangent line to a circle = No idea whether this will work or not, but hey it's a sandbox, won't hurt to try

From a given point outside a given circle, it is possible to draw a tangent to that circle.

Proof: WLOG, let the circle in question be the unit circle centered at the origin. The equation of the circle is then

$x^2 + y^2 = 1$

Let the arbitrary point be

$P_1 = (x_1,y_1)$

Case 1: The line in question is not vertical.

In that case, $x^2 + y^2 = 1$ represents an implicitly defined differentiable function of $x$. Taking the derivative WRT $x$ of both sides:

$2x + 2y \dfrac {\mathrm dy} {\mathrm dx} = 0$

$\dfrac {\mathrm dy} {\mathrm dx} = \dfrac {-x} {y}$

The tangent line, then, is of the form

$y - y_1 = m(x - x_1)$

And we will prove that this equation has roots, maybe. --GFauxPas 16:20, 14 November 2011 (CST)