Question

If $acos\theta +bsin\theta =4$ and $asin\theta -bcos\theta =3$, then $a^2+b^2=$

• A. 7
• B. 12
• C. 25
• D. 50

Answer 1

The correct option is C

25

$acos\theta +bsin\theta =4$ and $asin\theta -bcos\theta =3$
$\Rightarrow (acos\theta +bsin\theta {)}^2=4^{2}and(asin\theta -bcos\theta {)}^2=3^2$
Adding,
$a^{2}co{s}^2\theta +b^{2}si{n}^2\theta +2absin\theta cos\theta +a^{2}si{n}^2\theta +b^{2}co{s}^2\theta -2absin\theta cos\theta =16+9$
$a^2(co{s}^2\theta +si{n}^2\theta )+b^2(co{s}^2\theta +si{n}^2\theta )$ = 25

$\Rightarrow a^2+b^2=25$