Question

if abc=p where a,b and c belong to R  
and qa-b=0
then find the minimum distance of point (a,b,c) from the origin is
(answer in terms of p and q) 

Answer 1

qa-b=0
$a =\frac{b}{q}$
$abc= \frac{b^2c}{q}=p$
$b^2c=pq$
$b^2= \frac{p}{qc}$
similarly
$a^2= \frac{pq}{c}$
therefore let x be distance from origin
and let $d=x^2$
$d=a^2+b^2+c^2= \frac{pq}{c}+ \frac{p}{qc}+c^2$
$d'=2c- \frac{pq}{c^2}- \frac{p}{qc^2}$
if d'=0 then $c^3= \frac{p(q^2+1)}{2q}$
and u will also find that
$d"( \sqrt[3]{\frac{p(q^2+1)}{2q} } ) >0$
therefore
$d= x^{2} =3(\sqrt[3]{\frac{p(q^2+1)}{2q}})^2$
and so finally
$x= \sqrt{3} \sqrt[3]{\frac{p(q^2+1)}{2q}}$