Question

Solve for x and y : $12sinx+5cosx=2y^2-8y+21$.

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

$\sqrt{{12}^2+5^2}(\frac{12}{13}sinx+\frac{5}{13}cosx)=2(y^2-4y+4)+13$
or $13cos(x-a)=2(y-2{)}^2+13$,
where $cosa=\frac{5}{13}$
Clearly , LHS $\le$ 13 because the greatest value of cos(x - a) is 1 when x = a.
Also RHS $\ge$ 13 because the least value of RHS is 13 when y = 2
$\therefore$ the equation can hold if the value of each side = 13
Thus cos(x - a) = 1 and y = 2
x - a = 2n$\pi$ and y = 2
x = 2n$\pi$ + a and y = 2

$x=2n\pi +co{s}^{-1}\left(\frac{5}{13}\right)$ and y=2 where $n\in I$