Final answer:
To express the given equation in spherical coordinates, substitute x, y, and z with their respective spherical coordinates r, θ, and φ. However, the equation does not have a valid solution in spherical coordinates.
Step-by-step explanation:
The given equation is z = -sqrt(1/8(x^2 + y^2)). To express this equation in spherical coordinates, we need to substitute the rectangular coordinates, x, y, and z, with their respective spherical coordinates, r, θ (theta), and φ (phi). The relationship between spherical and rectangular coordinates is given by: x = r sin(θ) cos(φ), y = r sin(θ) sin(φ), and z = r cos(θ).
We will use these equations to express x, y, and z in terms of r, θ, and φ, and then substitute them into the given equation. After simplifying, we get the spherical coordinate equation: 0 = -sqrt(1/8(r^4 sin^2(θ) cos^2(φ) + r^4 sin^2(θ) sin^2(φ))). Since the square root of a negative number is not defined, there is no valid solution for this equation.