Question

if u²+v² varies as x²+y² and uv varies as xy, then prove that u+v varies as x+y when u/v + v/u = x/y + y/x

Answer 1

We recall the concept of proportionality

When quantities have same ratio they are proportional

Given:

$u^{2}+v^{2} = k_{1} [x^{2} +y^{2} ]$......................(1)

$uv=k_{2}xy$

$(u+v)^{2}=u^{2} +v^{2} +2uv$

             $=k_{1}(x^{2} +y^{2}) +2k_{2} xy$...............................(2)

   $\frac{u}{v}+\frac{v}{u}=\frac{x}{y} +\frac{y}{x}$

   $\frac{u^{2}+v^{2} }{uv} =\frac{x^{2}+y^{2} }{xy}$

$\frac{k_{1}(x^{2} +y^{2}) }{k_{2}x y} = \frac{(x^{2} +y^{2}) }{x y}$

           $k_{1}= k_{2}$

Substituting in (2),

$(u+v)^{2}=k_{1}(x+y)^{2}$

Hence, $u+v$ varies as $x+y$