Question

Find the standard form of the equation of the parabola with a focus at (0, -2) and a directrix at y = 2.

Answer 1

You can utilize the definition of parabola in order to find the equation of it. This definition is:
"The parabola is the geometric body defined by the set of point equidistant from one ficed point called Foci and a line called Directrix".
So, therefore, having a focus \(F( \alpha, \beta)\) and a directrix \(d)y=k\) then we will generalize a point \(M(x,y)\) and by definition, the distance from F to M must be the same as M to the directrix d:
\[dist (F,M)=dist(M,d)\]
Now, let's consider the directrix being unidirectional, meaning that we will draw a perpendicular line until it connects with M.
Therefore:
\[\sqrt{(x- \alpha)^2+(y-\beta)^2}=\left| y-k \right|\]
Simplifying further:
\[(x-\alpha)^2+(y-\beta^2)=(y-k)^2\]
And now simplifying firther you will obtain a parabola with the equation:
\[y=ax^2+bx+c\]