# Write The Equation In Spherical Coordinates. (A) X^{2} + Y^{2} + Z^{2} - 81 = 0 (B) X^{2 }- Y^{2 }- Z^{2 }- 1 = 0

**Solution:**

To convert cartesian into spherical coordinates we have to rely on the following relations:

r = ρsin𝝓 x = rcos θ = ρsin𝝓cos θ

z = ρcos𝝓 y = rsinθ = ρsin𝝓sin θ

The diagram from which the above relationships are derived is:

ρ = \(\sqrt{x^{2} + y^{2} + z^{2} } = \sqrt{r^{2} + z^{2}}\)

A.We use the relationships above to substitute for x, y, z and we get

ρ^{2}sin^{2}𝝓cos^{2}θ + ρ^{2}sin^{2}𝝓sin^{2}θ + ρ^{2}cos^{2}𝝓 = 81

ρ^{2}sin^{2}𝝓(cos^{2}θ + sin^{2}θ) + ρ^{2}cos^{2}𝝓 = 81

We know the fundamental relationship cos^{2}θ + sin^{2}θ = 1

ρ^{2}sin^{2}𝝓(1) + ρ^{2}cos^{2}𝝓 = 81

ρ^{2}[sin^{2}𝝓 + cos^{2}𝝓] = 81

Again sin^{2}𝝓 + cos^{2}𝝓 = 1, therefore

ρ^{2}[1] = 81

ρ^{2} = 81

**ρ = ± 9**

(B) X^{2 }- Y^{2 }- Z^{2 }- 1 = 0

r = ρsin𝝓 x = rcos θ = ρsin𝝓cos θ

z = ρcos𝝓 y = rsin θ = ρsin𝝓sin θ

X^{2 }- Y^{2 }- Z^{2} = 1

( ρsin𝝓cos θ )^{2} - (ρsin𝝓sin θ)^{2} - (ρcos𝝓 )^{2} =1

ρ^{2}sin^{2}𝝓cos^{2}θ - ρ^{2}sin^{2}𝝓sin^{2}θ - ρ^{2}cos^{2}𝝓 = 1

ρ^{2}sin^{2}𝝓(cos^{2}θ - sin^{2}θ) - ρ^{2}cos^{2}𝝓 = 1

ρ^{2}sin^{2}𝝓(cos^{2}θ - (1 - cos^{2}θ)) - ρ^{2}cos^{2}𝝓 = 1

ρ^{2}sin^{2}𝝓(2cos^{2}θ -1) - ρ^{2}cos^{2}𝝓 =1

2ρ^{2}sin^{2}𝝓cos^{2}θ - ρ^{2}sin^{2}𝝓 - ρ^{2}cos^{2}𝝓 = 1

2ρ^{2}sin^{2}𝝓cos^{2}θ - ρ^{2}[sin^{2}𝝓 + cos^{2}𝝓] = 1

Since sin^{2}𝝓 + cos^{2}𝝓 = 1

2ρ^{2}sin^{2}𝝓cos^{2}θ - ρ^{2}[1] = 1

ρ^{2}(2sin^{2}𝝓cos^{2}θ - 1) = 1

ρ^{2} = 1/(2sin^{2}𝝓cos^{2}θ - 1)

**ρ = 1/√(2sin ^{2}𝝓cos^{2}θ - 1)**

## Write The Equation In Spherical Coordinates. (A) X^{2} + Y^{2} + Z^{2} - 81=0 (B) X^{2 }- Y^{2 }- Z^{2 }- 1 = 0

**Summary:**

The equations (A) X^{2} + Y^{2} + Z^{2} - 8 1 = 0 (B) X^{2 }- Y^{2 }- Z^{2 }- 1 = 0 are represented in spherical coordinates as ρ = ± 9 and ρ = 1/√(2sin^{2}𝝓cos^{2} θ - 1) respectively.

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