Conic Sections Checking my solution: My Foci seams to be bigger than my Vertices (I think there is something wrong with my solution)

Question

Which one is my error? Pls help!

Answer 1

You have two mistakes.

$(x+1)^2+(2y-4)^2 = 4$ should be

$\displaystyle \small \frac{(x+1)^2}{4}+\frac{(y-2)^2}{1} = 1$

$c^2 = a^2-b^2 = 3 \implies c = \pm \sqrt3$

You also write $c^2 = a^2+b^2$ but that is not correct.

Given the center, the foci will be $(-1 \pm \sqrt3, 2)$

Answer 2

Note that

$$(2y-4)^2 \ne 2(y-2)^2$$

It should be

$$(2y-4)^2=4(y-2)^2$$