need help to understand answer

Question

Write the equation of a parabola with a vertex at $(-5, 2)$ and a directrix $y = -1$.

i got $(y-2)= \frac{1}{4} (x+5)^2$

Correct answer is $(y-2) = \frac{1}{12} (x+5)^2$

Answer 1

a parabola is the locus of a point which is equidistant from a line (the directrix) and a point not on the directrix (the focus).

since the vertex is at a distance $3$ from the directrix, the $y$-coordinate of the focus must be $2+3$, i.e. $5$. the $x$-coordinate in this case is the same as for the vertex, $-5$. Y take a point $(x,y)$ on the parabola:

distance from directrix = $y-(-1) = y + 1$

distance from focus = $\sqrt{(x-(-5))^2+(y-5)^2}$

squaring these distances and setting the results equal gives:

$$ (y+1)^2 = (x+5)^2+(y-5)^2 $$ rewriting as $$ (y+1)^2-(y-5)^2 = (x+5)^2+ $$ using the difference-of-two-squares factorization gives $$ 6(2y-4)=(x+5)^2 $$ which is easily reduced to the answer you require