# Write the equation in spherical coordinates. x^{2} + y^{2} + z^{2 }= 25.

**Solution:**

Given, the cartesian equation is x^{2} + y^{2} + z^{2} = 25.

We have to write the equation in spherical coordinates.

By using the relation to convert cartesian to spherical coordinates,

r = ρsin๐

x = rcos θ = ρsin๐cos θ

z = ρcos๐

y = rsinθ = ρsin๐sin θ

ρ = √x^{2} + y^{2} + z^{2} = √r^{2} + z^{2}

Now, x^{2} + y^{2} + z^{2} = r^{2} + z^{2}

According to the question,

(ρsin๐cos θ)^{2} + (ρsin๐sinθ) + (ρcos๐)^{2} = 25

ρ^{2}sin^{2}๐cos^{2}θ + ρ^{2}sin^{2}๐sin^{2}θ + ρ^{2}cos^{2}๐ = 25

Taking out common term,

ρ^{2}sin^{2}๐(cos^{2}θ + sin^{2}θ) + ρ^{2}cos^{2}๐ = 25

We know, cos^{2}θ + sin^{2}θ = 1

ρ^{2}sin^{2}๐(1) + ρ^{2}cos^{2}๐ = 25

Taking out common term,

ρ^{2}[sin^{2}๐ + cos^{2}๐] = 25

Also, sin^{2}๐ + cos^{2}๐ = 1,

ρ^{2}[1] = 25

ρ^{2} = 25

ρ = ±5

X^{2} - Y^{2} - Z^{2} - 1 = 0

r = ρsin๐ x = rcos θ = ρsin๐cos θ

z = ρcos๐ y = rsin θ = ρsin๐sin θ

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

( ρsin๐cos θ)^{2} - (ρsin๐sin θ)^{2} - (ρcos๐)^{2} =1

ρ^{2}sin^{2}๐cos^{2}θ - ρ^{2}sin^{2}๐sin^{2}θ - ρ^{2}cos^{2}๐ = 1

ρ^{2}sin^{2}๐(cos^{2}θ - sin^{2}θ) - ρ^{2}cos^{2}๐ = 1

ρ^{2}sin^{2}๐(cos^{2}θ - (1 - cos^{2}θ)) - ρ^{2}cos^{2}๐ = 1

ρ^{2}sin^{2}๐(2cos^{2}θ -1) - ρ^{2}cos^{2}๐ =1

2ρ^{2}sin^{2}๐cos^{2}θ - ρ^{2}sin^{2}๐ - ρ^{2}cos^{2}๐ = 1

2ρ^{2}sin^{2}๐cos^{2}θ - ρ^{2}[sin^{2}๐ + cos^{2}๐] = 1

Since sin^{2}๐ + cos^{2}๐ = 1

2ρ^{2}sin^{2}๐cos^{2}θ - ρ^{2}[1] = 1

ρ^{2}(2sin^{2}๐cos^{2}θ - 1) = 1

ρ^{2} = 1/(2sin^{2}๐cos^{2}θ - 1)

ρ = 1/√(2sin^{2}๐cos^{2}θ - 1)

Therefore, the equation in spherical coordinates is ρ = 1/√(2sin^{2}๐cos^{2}θ - 1).

## Write the equation in spherical coordinates. x^{2} + y^{2} + z^{2 }= 25.

**Summary:**

The equation in spherical coordinates. x^{2} + y^{2} + z^{2} = 25 is ρ = 1/√(2sin^{2}๐cos^{2}θ - 1).

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