If $\frac{3x+2iy}{5i-2}=\frac{15}{8x+3iy}$, then

Given that $\frac{3x+2iy}{5i-2}=\frac{15}{8x+3iy}$

$\Rightarrow 24x^2+9ixy-6y^2+16ixy=75i-30$

$\Rightarrow 24x^2-6y^2+25ixy=75i-30$

Equating real and imaginary parts, we get

$24x^2-6y^2=-30 or 4x^2-y^2=-5$ and $xy=3$

On solving we get $x=\pm 1,y=\pm 3$