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A day full of math games & activities. Find one near you.

Write the equation in spherical coordinates. x2 + y2 + z2 = 49.

Solution:

Given, the cartesian equation is x2 + y2 + z2 = 49.

We have to write the equation in spherical coordinates.

By using the relation to convert cartesian to spherical coordinates,

r = ρsin๐“ 

x = rcos θ = ρsin๐“cos θ 

z = ρcos๐“ 

y = rsinθ = ρsin๐“sin θ

Write the equation in spherical coordinates. x2 + y2 + z2 = 49

 ρ = √x2 + y2 + z2 = √r2 + z2

Now, x2 + y2 + z2 = r2 + z2

According to the question,

(ρsin๐“cos θ)2 + (ρsin๐“sinθ) + (ρcos๐“)2 = 49

ρ2sin2๐“cos2θ + ρ2sin2๐“sin2θ + ρ2cos2๐“ = 49

Taking out common term,

ρ2sin2๐“(cos2θ + sin2θ) + ρ2cos2๐“ = 49

We know, cos2θ + sin2θ = 1

ρ2sin2๐“(1) + ρ2cos2๐“ = 49

Taking out common term,

ρ2[sin2๐“ + cos2๐“] = 49

Also, sin2๐“ + cos2๐“ = 1,

ρ2[1] = 49

ρ2 = 49

ρ = ±7

X2 - Y2 - Z2 - 1 = 0

r = ρsin๐“ x = rcos θ = ρsin๐“cos θ 

z = ρcos๐“ y = rsin θ = ρsin๐“sin θ

 X2 - Y2 - Z2 = 1

(ρsin๐“cos θ)2 - (ρsin๐“sin θ)2 - (ρcos๐“)2 =1

ρ2sin2๐“cos2θ - ρ2sin2๐“sin2θ - ρ2cos2๐“ = 1

ρ2sin2๐“(cos2θ - sin2θ) - ρ2cos2๐“ = 1

ρ2sin2๐“(cos2θ - (1 - cos2θ)) - ρ2cos2๐“ = 1

ρ2sin2๐“(2cos2θ -1) - ρ2cos2๐“ =1 

2ρ2sin2๐“cos2θ - ρ2sin2๐“ - ρ2cos2๐“ = 1

2ρ2sin2๐“cos2θ - ρ2[sin2๐“ + cos2๐“] = 1

Since sin2๐“ + cos2๐“ = 1

2ρ2sin2๐“cos2θ - ρ2[1] = 1

ρ2(2sin2๐“cos2θ - 1) = 1

ρ2 = 1/(2sin2๐“cos2θ - 1)

ρ = 1/√(2sin2๐“cos2θ - 1)

Therefore, the equation in spherical coordinates is ρ = 1/√(2sin2๐“cos2θ - 1).

Write the equation in spherical coordinates. x2 + y2 + z2 = 49.

Summary:

The equation in spherical coordinates x2 + y2 + z2 = 49 is ρ = 1/√(2sin2๐“cos2θ - 1).

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