Question

If $x=rsin\alpha cos\beta ,y=rsin\alpha sin\beta$ and $z=rcos\alpha$, then

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

Answer 1

The correct option is A

$x^2+y^2+z^2=r^2$

Given:

$x=rsin\alpha cos\beta ,y=rsin\alpha sin\beta andz=rcos\alpha \therefore x^2+y^2+z^2=r^{2}si{n}^2\alpha co{s}^2\beta +r^{2}si{n}^2\alpha si{n}^2\beta +r^{2}co{s}^2\alpha =r^{2}si{n}^2\alpha (co{s}^2\beta +si{n}^2\beta )+r^{2}co{s}^2\alpha =r^{2}si{n}^2\alpha +r^{2}co{s}^2\alpha =r^2$

$\Rightarrow x^2+y^2+z^2=r^2$