how do you solve this calculus question?

Question

Find the points on the curve:

x^2 + xy + y^2 = 7

A) where the tangent is parallel to the x axis

B) where the tangent is parallel to the y axis

Answer 1

2xdx + ydx + xdy + 2ydy = 0
dy/dx = - (y + 2x)/(2y + x)
A) y = - 2x
x^2 - 2x^2 + 4x^4 = 7
4x^4 - x^2 - 7 = 0
x^4 - (1/4)x^2 - 7/4 = 0
x^4 - (1/4)x^2 + 1/64 - 1/64 - 7/4 = 0
(x^2 - 1/8)^2 - (1 + 112)/64 = 0
(x^2 - 1/8)^2 - 113/64 = 0
(x^2 - 1/8(1 + √113) (x^2 - 1/8(1 - √113) = 0
Discarding the two complex roots,
(x - √(1/8(1 + √113)) (x + √(1/8(1 + √113)) = 0
x = ± √(1/8(1 + √113)
y^2 ± y√(1/8(1 + √113) + 1/8(1 + √113) = 7
y^2 ± y√(1/8(1 + √113) - 1/8(55 - √113) = 0
y^2 ± y√(1/8(1 + √113) + (1/4)(1/8(1 + √113) + (1/4)(1/8(1 + √113) - 1/8(55 - √113) = 0
(y ± (1/2)√(1/8(1 + √113))^2 - (1/8)((59/4) - (3/4)√113) = 0
y = ± (1/2)√(1/8(1 + √113) ± √(1/8)((59/4) - (3/4)√113)
B)
Substitute y for x in the above

Answer 2

Use implicit differentiation and solve for dy/dx and then find where its 0 (the answer to A) and undefined (the answer to B)