Question

If $$a : b :: c : d$$, show that $$\frac{ma^2 + nc^2}{mb^2 + nd^2} = \frac{\sqrt{a^4 + c^4}}{\sqrt{b^4 + d^4}}$$.

Answer 1

Let $$\frac{a}{b} = \frac{c}{d} = k$$, so $$a = bk$$ and $$c = dk$$.

$$\text{LHS} = \frac{m(b^2k^2) + n(d^2k^2)}{mb^2 + nd^2}$$

$$= \frac{k^2(mb^2 + nd^2)}{mb^2 + nd^2}$$

$$= k^2$$

$$\text{RHS} = \frac{\sqrt{b^4k^4 + d^4k^4}}{\sqrt{b^4 + d^4}}$$

$$= \frac{k^2\sqrt{b^4 + d^4}}{\sqrt{b^4 + d^4}}$$

$$= k^2$$

$$\text{LHS} = \text{RHS}$$