MCQ
If $\text{x}=\text{r}\sin\theta\cos\phi,\text{y}=\text{r}\sin\phi$ and ${z}=\text{r}\cos\theta,$ then :
- ✓$x^2+y^2+z^2=r^2 $
- B$ x^2+y^2-z^2=r^2 $
- C$ x^2-y^2+z^2=r^2 $
- D$ z^2+y^2-x^2=r^2 $
✓
Answer
Correct option: A.
$x^2+y^2+z^2=r^2 $
$\text{x}=\text{r}\sin\theta\cos\phi$
$\text{y}=\text{r}\sin\theta\sin\phi$
$\text{z}=\text{r}\cos\theta$
Squaring and adding these equations, we get
$\text{x}^2+\text{y}^2+\text{z}^2=(\text{r}\sin\theta\cos\phi)^2+(\text{r}\sin\theta\sin\phi)^2+(\text{r}\cos\theta)^2$
$\Rightarrow\text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2\sin^2\theta\cos^2\phi+\text{r}^2\sin^2\theta\sin^2\phi+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=(\text{r}^2\sin^2\theta\cos^2\phi+\text{r}^2\sin^2\theta\sin^2\phi)+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2\sin^2\theta(\cos^2\phi+\sin^2\phi)+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2\sin^2\theta(1)+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2\sin^2\theta+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2(\sin^2\theta+\cos^2\theta)$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2(1)$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2$
Hence, the correct option is $(A).$
$\text{y}=\text{r}\sin\theta\sin\phi$
$\text{z}=\text{r}\cos\theta$
Squaring and adding these equations, we get
$\text{x}^2+\text{y}^2+\text{z}^2=(\text{r}\sin\theta\cos\phi)^2+(\text{r}\sin\theta\sin\phi)^2+(\text{r}\cos\theta)^2$
$\Rightarrow\text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2\sin^2\theta\cos^2\phi+\text{r}^2\sin^2\theta\sin^2\phi+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=(\text{r}^2\sin^2\theta\cos^2\phi+\text{r}^2\sin^2\theta\sin^2\phi)+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2\sin^2\theta(\cos^2\phi+\sin^2\phi)+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2\sin^2\theta(1)+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2\sin^2\theta+\text{r}^2\cos^2\theta$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2(\sin^2\theta+\cos^2\theta)$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2(1)$
$\Rightarrow\ \text{x}^2+\text{y}^2+\text{z}^2=\text{r}^2$
Hence, the correct option is $(A).$
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