Question

If $a\cos \theta -b\sin \theta =c$,then $a\sin \theta +b\cos \theta =$

• A. $\pm \sqrt{a^2+b^2+c^2}$
• B. $\pm \sqrt{a^2+b^2-c^2}$
• C. $\pm \sqrt{c^2-a^2-b^2}$
• D. None of these

Answer 1

The correct option is B $\pm \sqrt{a^2+b^2-c^2}$

Given $a\cos \theta -b\sin \theta =c$ ........(i)
now let $a\sin \theta +b\cos \theta =k$ ........(ii)
by squaring equation (i) and (ii) and then adding
$a^2({\sin }^2\theta +{\cos }^2\theta )+b^2({\sin }^2\theta +{\cos }^2\theta )=c^2+k^2$
$a^2+b^2=c^2+k^2$
$k^2=a^2+b^2-c^2$
$k=\pm \sqrt{a^2+b^2-c^2}$