I have this MCQ.
Which of the following is the equation of the parabola with focus at $(1,2)$ and vertex at $(3,2)$ :
A. $ y^2 - 4y + 8x - 20 = 0 $
B. $ y^2 + y + 8x -20 = 0 $
C. $ y^2 + 4y + 8x - 20 = 0 $
D. $ y^2 - 4y + x - 20 = 0 $
E. $ y^2 - 4y + 8x + 20 = 0 $
Now, I know how to find equation of a parabola from focus and vertex, or from directrix and vertex, but that takes some time. I was wondering if there is a way to just sort of see which equation is the right one, because right from the focus and vertex we can see that the parabola will open on the left and its equation will be $y^2 = - 4ax$.
I'm not sure if it's actually possible though, but I think it would be cool if there was a way.
You don't need any particular formula here: just plug the coordinates of the vertex into the equation and you'll see that only in case A the point belongs to the parabola.