Question:
Find the equation of the parabola whose focus is $(1, -1)$ and whose vertex is $(2, 1)$.
My answer: $$\left(x-1\right)^2+\left(y+1\right)^{2\ }=\frac{\left(2y\ +x\ -9\right)^2}{5}$$
Answer key:
$$\left(2x\ -\ y\ -3\right)^{2\ }=-20\left(x+2y\ -4\right)$$
Discussion:
It might initially appear that the two equations are different; however, they are identical.
My solution was to find the equation of the directrix. The idea was to find the distance between a variable point — $(x,y)$ — on the parabola from the directrix and the focus. The distances must be equal.
The interesting part is that the solution given in the answer key takes the following form:
$$(\text{axis})^2 = -20(\text{line passing through the vertex and perpendicular to the axis})$$
Is the above form always true? If the above form has a geometrical meaning, what quantity does the $-20$ indicate?
Graphs:
