Q. 194.1( 9 Votes )
If x = r sin <spa A. x2 + y2 + z2 = r2
B. x2 + y2− z2 = r2
C. x2− y2 + z2 = r2
D. z2 + y2− x2 = r2
Answer :
Given: x = r sin θ cos ϕ, y = r sin θ sin φ and z = r cos θ,
Solution:
x = r sin θ cos ϕ
Squaring both sides, we get
x2 = r2 sin2 θ cos2ϕ ……….(i)
and y = r sin θ sin ϕ
Squaring both sides, we get⇒ y2 = r2 sin2 θ sin2ϕ ……….(ii)
z = r cos θ
Squaring both sides, we get
⇒ z2 = r2 cos2 θ ……….(iii)
Adding (i), (ii) and (iii), we get
x2 + y2 + z2 = r2 sin2 θ cos2ϕ + r2 sin2 θ sin2ϕ + r2 cos2 θ
= r2 (sin2 θ cos2ϕ + sin2 θ sin2ϕ + cos2 θ)
= r2 [sin2 θ (cos2ϕ + sin2ϕ) + cos2 θ]
∵ sin2 θ + cos2 θ = 1= r2 [sin2 θ + cos2 θ]
Again apply the identity sin2 θ + cos2 θ = 1= r2
Hence x2 + y2 + z2 = r2
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