Talk:Equation of Straight Line Tangent to Circle

Point out here that $(x_n, y_n)$ is actually on the circle. Or can this be result be expanded to the general case where $(x_n, y_n)$ is outside the circle (two lines) and prove that there are no tangents to a point inside? - Prime.mover

I tried to work out a proof for this and asked my professor about it. She said that it can be used to prove that there are no tangents to a point inside (edit: and that there is a solution for every point outside) but she hasn't taught me how yet, and will get to it in a few months. What I have to do is prove that all the solutions to $\frac {\mathrm dy}{\mathrm dx}$ are on the $xy$-plane outside of the circle. I imagined a tangent line moving about the circle and filling in all the space that it crosses, and intuitively the filled space is everywhere on the plane outside the circle. --GFauxPas 09:55, 22 November 2011 (CST)

Should be high-school stuff but never mind (you get a square root of a negative number). Never mind. --prime mover 13:17, 22 November 2011 (CST)

It uses basic algebra of quadratic equations which should be within the grasp of high-school mathematics. Take the equation of a circle: $C: (x-a)^2 + (y-b)^2 = c^2$, and the point $P = (x_1, y_1)$ or whatever, and find an equation for the lines from $P$ tangent to $C$, and it should be straightforward to get a quadratic equation which has 2 solutions when $P$ is outside, 1 when on, and 0 when inside the circle. I'll have a go myself in a bit, but I'm tired. --prime mover 14:43, 22 November 2011 (CST)