Final answer:
To find the points on the given surface where the tangent plane is parallel to the xy-plane, we need to find where the z-coordinate is constant. By taking the partial derivatives of the equation with respect to x and y and setting them equal to zero, we find four points: (2, 2, 2), (1, 3, 0), (1, -3, 0), and (-1, -3, 0).
Step-by-step explanation:
The equation given is 5x² + 4y² + 1z² = 1.
To find the points on this surface at which the tangent plane is parallel to the xy-plane, we need to find the points where the z-coordinate is constant. In other words, we need to find the points where dz/dx = 0 and dz/dy = 0.
By taking partial derivatives of the equation with respect to x and y, we get:
d/dx(5x² + 4y² + z²) = 10x
d/dy(5x² + 4y² + z²) = 8y
Setting these equal to zero and solving for x and y, we find four points: (2, 2, 2), (1, 3, 0), (1, -3, 0), and (-1, -3, 0).