Find points when tangent to ellipse is horizontal

I have this question...

[A] The cone with equation and the plane with equation intersect in an ellipse. Write an equation of the plane normal to this ellipse at the point P(3, 4, -5).

[B] It is apparent from geometry that the highest and lowest points of the ellipse from [A] are those points where its tangent line is horizontal. Find those points.

For part [A], I took the gradient vectors of the two functions and used their cross product to find the ellipse . I know this answer is correct, but I don't have an answer to part B to check against. Any insight for part B would be greatly appreciated!

Thanks!