If a/b = c/d = e/f prove that: (i) \[({{\mathbf{b}}^{\mathbf{2}}}~+\text{ }{{\mathbf{d}}^{\mathbf{2}}}~+\text{ }{{\mathbf{f}}^{\mathbf{2}}})\text{ }({{\mathbf{a}}^{\mathbf{2}}}~+\text{ }{{\mathbf{c}}^{\mathbf{2}}}~+\text{ }{{\mathbf{e}}^{\mathbf{2}}})\text{ }=\text{ }{{\left( \mathbf{ab}\text{ }+\text{ }\mathbf{cd}\text{ }+\text{ }\mathbf{ef} \right)}^{\mathbf{2}}}\] (ii) \[\frac{{{({{a}^{3}}+{{c}^{3}})}^{2}}}{{{({{b}^{3}}+{{d}^{3}})}^{2}}}=\frac{{{e}^{6}}}{{{f}^{6}}}\]

Consider

a/b = c/d = e/f = k

So we get

a = bk, c = dk, e = fk

(i) LHS = \[({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }({{a}^{2}}~+\text{ }{{c}^{2}}~+\text{ }{{e}^{2}})\]

We can write it as

\[=\text{ }({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }({{b}^{2}}{{k}^{2}}~+\text{ }{{d}^{2}}{{k}^{2}}~+\text{ }{{f}^{2}}{{k}^{2}})\]

Taking out the common terms

\[=\text{ }({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\text{ }{{k}^{2}}~({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\]

So we get

\[=\text{ }{{k}^{2}}~({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})\]

RHS \[=\text{ }{{\left( ab\text{ }+\text{ }cd\text{ }+\text{ }ef \right)}^{2}}\]

We can write it as

\[=\text{ }{{\left( b.\text{ }kb\text{ }+\text{ }dk.\text{ }d\text{ }+\text{ }fk.\text{ }f \right)}^{2}}\]

So we get

\[=\text{ }(k{{b}^{2}}~+\text{ }k{{d}^{2}}~+\text{ }k{{f}^{2}})\]

Taking out common terms

\[=\text{ }{{k}^{2}}~{{({{b}^{2}}~+\text{ }{{d}^{2}}~+\text{ }{{f}^{2}})}^{2}}\]

Therefore, LHS = RHS.

Therefore, LHS = RHS