Question

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

Answer 1

`y=-1/2x^2-x`

Explanation:

Parabola is the locus of a point, say `(x,y)`, 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=ax^2+bx+c`

As focus is `(-1,18)`, distance of `(x,y)` from it is `sqrt((x+1)^2+(y-18)^2)`

and distance of `(x,y)` from directrix `y=19` is `(y-19)`

Hence equation of parabola is

`(x+1)^2+(y-18)^2=(y-19)^2`

or `(x+1)^2=(y-19)^2-(y-18)^2=(y-19-y+18)(y-19+y-18)`

or `x^2+2x+1=-1(2y-1)=-2y+1`

or `2y=-x^2-2x`

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