If the focus of the parabola is (α,β) and the equation of the directrix is y=mx+c, find the equation of the parabola.

Question

Thanks a lot in advance for any help. I've completely forgotten conic sections over time, and need some help revising them.

Answer 1

Due to the definition of a parabola, (distance MF=distance MH to the directrix, set to the square), the second degree equation is

$$\tag{1}(x-\alpha)^2+(y-\beta)^2=\frac{(y-mx-c)^2}{1+m^2}$$

Explanation : we use the following formula for the distance from $(x_0,y_0)$ to straight line with equation $ax+by+c=0$ :

$$d=\dfrac{|ax+by+c|}{\sqrt{a^2+b^2}}$$

This formula is established in (http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html).