Can you help me come up with an equation for a parabola with vertex at (-2,4) and focus at (-4,1) ?

Question

I am stuck on this problem because I am not sure how to rotate the parabola. I used the equation x=a(y-k)^2+h with (h,k)=(-2,4). I think this problem has something to do with rotation but I'm not sure how to go about it. I appreciate the help!

Answer 1

one method is to find a vertical or horizontal parabola equation with the same vertex

such as y = a(x+2)^2 +4

find another point on the parabola to solve for a

then convert the rectangular equation into a polar equation.

where r^2 = x^2 +y^2 and tangent of the angle = y/x. The change the angle to rotate the parabola.

then convert back to rectangular coordinates.

the calculations get very messy.