Question

The value of $a$ for which system of equations ${\sin }^{2}x+{\cos }^{2}y=\frac{3a}{2}$ and ${\cos }^{2}x+{\sin }^{2}y=\frac{a^2}{2}$ has a solution is

Answer 1

Given system of equations
${\sin }^{2}x+{\cos }^{2}y=\frac{3a}{2}$ ....(1)
and ${\cos }^{2}x+{\sin }^{2}y=\frac{a^2}{2}$ ....(2)
Adding (1) and (2), we get
$1+1=\frac{3a}{2}+\frac{a^2}{2}$
$\Rightarrow a^2+3a-4=0$
$\Rightarrow a=-4,a=1$
Here $a=-4$ is not possible as LHS of equation (1) is always positive.
So, $a=1$