# Thread: can parabolas have tangent lines?

1. ## can parabolas have tangent lines?

Prove that the two tangents to a parabola from any point on the directix are perpendicular.

Im not sure about this question. I was drawing it out and like.. can you have a line thats tangent to a parabola or wont they all eventually cross it? I think if yes... then wouldnt it be more then 2 tangent lines ? anyway can anyone maybe talk about this problem and maybe point me in the right direction?

2. A parabola is the set of all points equidistant from a point (the focus) and a line (the directrix). In this problem, you're asked to pick an arbitrary point on the directrix, find the equations for the two tangent lines to the parabola that go through the arbitrary point on the directrix, and show that the slopes of those two lines are negative inverses of each other. That is, if $\displaystyle P$ is a point on the directrix, and line $\displaystyle 1$ is tangent to the parabola at some point and also contains $\displaystyle P$ and has slope $\displaystyle m_{1},$ and if it's also true that line $\displaystyle 2$ is tangent to the parabola at some point and also contains point $\displaystyle P$ and has slope $\displaystyle m_{2},$ then you're asked to show that $\displaystyle m_{1}=-1/m_{2}.$

Does that make sense?

3. Originally Posted by emakarov
A picture is worth a thousand words.