Answer:
[tex]\mathrm{Parabola\:focus\:given}\:x^2=-20y:\quad \left(0,\:-5\right)[/tex]
Step-by-step explanation:
Given the equation
[tex]x2 = -20y[/tex]
A parabola is the locus of points such that the distance to a point the focus equals the distance to a line the directrix.
[tex]4p\left(y-k\right)=\left(x-h\right)^2[/tex] is the standard equation for an up-down facing parabola with vertex at (h, k), and a focal length |p|.
so
[tex]x^2=-20y[/tex]
[tex]\mathrm{Switch\:sides}[/tex]
[tex]-20y=x^2[/tex]
[tex]\mathrm{Factor\:}4[/tex]
[tex]4\cdot \frac{-20}{4}y=x^2[/tex]
[tex]4\left(-5\right)y=x^2[/tex]
[tex]\mathrm{Rewrite\:as}[/tex]
[tex]4\left(-5\right)\left(y-0\right)=\left(x-0\right)^2[/tex]
[tex]\left(h,\:k\right)=\left(0,\:0\right),\:p=-5[/tex]
Parabola is symmetric around the y-axis and so the focus lies a distance\ p from the center (0, 0) along the y-axis.
[tex]\left(0,\:0+p\right)[/tex]
[tex]=\left(0,\:0+\left(-5\right)\right)[/tex]
[tex]\mathrm{Refine}[/tex]
[tex]\left(0,\:-5\right)[/tex]
Therefore,
[tex]\mathrm{Parabola\:focus\:given}\:x^2=-20y:\quad \left(0,\:-5\right)[/tex]
Please check the attached figure too.
