If x -rsin-cos- y -rsin-sin- and z - r cos- then x 2- y 2- z 2 is independent ofA- 0- -B- c - -C- r - -D- r

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Basic Trigonometric Identities

If x =rsinθco...

Question

If $x = r s i n θ c o s ϕ, y = r s i n θ s i n ϕ$ and $z = r c o s θ$ , then $x^{2} + y^{2} + z^{2}$ is independent of

$0, ϕ$

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$r, θ$

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$c, ϕ$

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Solution

The correct option is A
$0, ϕ$

$0, ϕ$

We have: $x = r s i n θ c o s ϕ, y = r s i n θ s i n ϕ$ and z

$= r c o s θ,$

$∴ x^{2} + y^{2} + z^{2}$

$= (r s i n θ c o s ϕ)^{2} + (r s i n θ s i n ϕ)^{2} + (r c o s θ)^{2}$

$= r^{2} s i n^{2} θ c o s ϕ^{2} + r^{2} s i n^{2} θ s i n^{2} ϕ + r^{2} c o s^{2} θ$

$= r^{2} s i n 62 θ (c o s^{2} ϕ + s i n^{2} ϕ) + r^{2} c o s^{2} θ$

$= r^{2} s i n^{2} θ \times 1 + r^{2} c o s^{2} θ$

$= r^{2} s i n^{2} θ + r^{2} c o s^{2} θ$

$= r^{2} (s i n^{2} θ + c o s^{2} θ)$

$= r^{2} \times 1$

$= r^{2}$

Thus, $x^{2} + y^{2} + z^{2}$ is independent of $θ$ and $ϕ$ .

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Similar questions

If $x = r sin θ cos ϕ, y = r sin θ sin θ sin ϕ$ and $z = r cos θ$ , then $x^{2} + y^{2} + z^{2}$ is independent of
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(b)

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If x=rsinθcosϕ,y=rsinθsinϕ and z=rcosθ, then x2+y2+z2 is independent of

If $x = r s i n θ c o s ϕ, y = r s i n θ s i n ϕ$ and $z = r c o s θ$ , then $x^{2} + y^{2} + z^{2}$ is independent of