What is the equation in standard form of the parabola with a focus at (-1,18) and a directrix of y= 19?

Feb 8, 2017

$y = - \frac{1}{2} {x}^{2} - x$

Explanation:

Parabola is the locus of a point, say $\left(x , y\right)$, which moves so that its distance from a given point called focus and from a given line called directrix , is always equal.

Further, standard form of equation of a parabola is $y = a {x}^{2} + b x + c$

As focus is $\left(- 1 , 18\right)$, distance of $\left(x , y\right)$ from it is $\sqrt{{\left(x + 1\right)}^{2} + {\left(y - 18\right)}^{2}}$

and distance of $\left(x , y\right)$ from directrix $y = 19$ is $\left(y - 19\right)$

Hence equation of parabola is

${\left(x + 1\right)}^{2} + {\left(y - 18\right)}^{2} = {\left(y - 19\right)}^{2}$

or ${\left(x + 1\right)}^{2} = {\left(y - 19\right)}^{2} - {\left(y - 18\right)}^{2} = \left(y - 19 - y + 18\right) \left(y - 19 + y - 18\right)$

or ${x}^{2} + 2 x + 1 = - 1 \left(2 y - 1\right) = - 2 y + 1$

or $2 y = - {x}^{2} - 2 x$

or $y = - \frac{1}{2} {x}^{2} - x$
graph{(2y+x^2+2x)(y-19)=0 [-20, 20, -40, 40]}