Recalling the definition of a parabola:
Parabola: A parabola is a curve where any point is at an equal distance from
1) a fixed point (f₁,f₂) (the focus ), and
2) fixed straight line Ax+By+C=0 (the directrix).
Therefore, a point (x₀,y₀) is on the parabola iff:
[tex]\sqrt[]{(x_0-f_1)^2+(y_0_{}-f_2)^2}=|\frac{Ax_0+By_0+C}{\sqrt[]{A^2+B^2}}|[/tex]Now, if the directrix is y=3 and the focus is (0,-3), then (x₀,y₀) is on the parabola iff:
[tex]\sqrt[]{(x_0-0)^2+(y_0-(-3))^2}=|y_0-3|[/tex]Raising the equation to power 2 and solving for y₀ we get:
[tex]\begin{gathered} x^2_{0^{}}+(y_0+3)^2=(y_0-3)^2 \\ x^2_0+y^2_0+6y_0+9=y^2_0-6y_0+9 \\ x^2_0=-12y_0 \\ y_0=-\frac{x^2_0}{12} \end{gathered}[/tex]Therefore the equation of the parabola with focus (0,-3) and directrix y=3 is:
[tex]y=-\frac{x^2}{12}[/tex]
