Write the equation in spherical coordinates. (a) 4z2 = 5x2 + 5y2 (b) x2 + 3z2 = 4

Refer to the diagram shown below.

The transformation from the Cartesian coordinate system (x,y,z) to the spherical coordinate system (r,θ,φ) is given by
x = rsinφcosθ
y = rsinφsinθ
z = rcosφ

Part a.
The given equation is 4z² = 5x² + 5y².
In spherical coordinates, obtain
4r²cos²φ = 5r²sin²φ cos²θ + 5r²sin²φ sin²θ
             = 5r²sin²φ(cos²θ + sin²θ)
             = 5r²sin²φ
4/5 = tan²φ
tanφ = 2/√5 = (2/5)√5

  tanφ = (2/5)√5

Part b.
The given equation is x² + 3z² = 4
In spherical coordinates, obtain
r²sin²φ cos²θ + 3r²cos²φ = 4

  r²(sin²φ cos²θ + 3cos²φ) = 4

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