How to solve this system of conics?

Question

I am currently trying to figure out how to solve the following systems of conics:

$\frac{(x+1)^2}{16} + \frac{(y-1)^2}{81} = 1$

$x+6=\frac{1}{4}(y-1)^2$

How would I find the four points that these two equations intersect?

Thank you in advance.

Answer 1

A short approach would be-
$\frac{4(x+6)}{81}=\frac{(y-1)^2}{81}$
Now substituting in equation one, we get-
$\frac{(x+1)^2}{16}+\frac{4(x+6)}{81}=1$
It will seem quadratic at first, but for each x value, there are $2$ y-values. Now simply solve the equation.

Answer 2

Short answer.

Eliminate $(y-1)^2 $ you get two $x$ s.

Plug into the parabola equation and find two $y$ s.