Question

If $x^2+y^2+z^2=r^2$, where $x=r\sin A\cos B,y=r\sin A\sin B$, then z has one of the following values :

• A. $r\sin B$
• B. $r\cos A$
• C. $r\tan A\cos B$
• D. $r\tan A\tan B$

Answer 1

The correct option is B $r\cos A$

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

$\Rightarrow x^2+y^2=r^2-z^2$
$x^2=r^2{\sin }^{2}A{\cos }^{2}B,$
$y^2=r^2{\sin }^{2}A{\cos }^{2}B$
$r^2{\sin }^{2}A{.\cos }^{2}B+r^2{\sin }^{2}A.{\sin }^{2}B=r^2-z^2$
$r^2{\sin }^{2}A({\cos }^{2}B+{\sin }^{2}B)=r^2-z^2$
$r^2{\sin }^{2}A=r^2-z^2$
$\Rightarrow z^2=r^2-r^2{\sin }^{2}A=r^2(1-{\sin }^{2}A)$
$\Rightarrow z^2=r^2{\cos }^{2}A$

$\Rightarrow z=r\cos A$