Question

If $x=r\sin \theta \cos \varphi ,y=r\cos \theta \cos \varphi$ and $z=r\sin \varphi$, then _____.

• A. $x^2+y^2+z^2=r^2$
• B. $x^2+y^2-z^2=4r^2$
• C. $x^2-y^2+z^2=r^2$
• D. $x^2+y^2-z^2=r^2$

Answer 1

The correct option is A $x^2+y^2+z^2=r^2$

$x=r\sin \theta \cos \varphi y=r\cos \theta \cos \varphi z=r\sin \varphi$

Now,

$x^2=r^2{\sin }^2\theta {\cos }^2\varphi y^2=r^2{\cos }^2\theta {\cos }^2\varphi z^2=r^2{\sin }^2\varphi$

Adding the above three expressions we get,

$x^2+y^2+z^2=r^2{\sin }^2\theta {\cos }^2\varphi +r^2{\cos }^2\theta {\cos }^2\varphi +r^2{\sin }^2\varphi =r^2{\cos }^2\varphi ({\sin }^2\theta +{\cos }^2\theta )+r^2{\sin }^2\varphi =r^2{\cos }^2\varphi +r^2{\sin }^2\varphi =r^2({\cos }^2\varphi +{\sin }^2\varphi )=r^2$