Q2014/098) Given $a^2+b^2=16$, $c^2+d^2=25$ find the maximium of ac

without loss of generality we can choose
$a=4\sin\,t$
$b=4\cos\,t$
$c=5\sin\,p$
$d=5\cos\,p$


so we get $ad-bc= 20\sin\, t \cos\, p - 20\sin\, p \cos\, t = 20\sin (t-p) = 20$
or $\sin(t-p) = 1$
so $t= p+ \dfrac{\pi}{2}$
hence
$ac = 20 \sin \, t \sin \ p$
= $20 \sin\, p + \sin (\dfrac{\pi}{2}+ p)$
= $-20 \cos \, p \sin\, p$
= $-10 \sin 2p$


clearly the largest value is 10 and smallest -10