Question

If $$a : b :: c : d$$, show that $$\frac{a + b}{c + d} = \frac{\sqrt{2a^2 + 7b^2}}{\sqrt{2c^2 + 7d^2}}$$.

Answer 1

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

$$\text{LHS} = \frac{bk + b}{dk + d}$$

$$= \frac{b(k + 1)}{d(k + 1)}$$

$$= \frac{b}{d}$$

$$\text{RHS} = \frac{\sqrt{2b^2k^2 + 7b^2}}{\sqrt{2d^2k^2 + 7d^2}}$$

$$= \frac{b\sqrt{2k^2 + 7}}{d\sqrt{2k^2 + 7}}$$

$$= \frac{b}{d}$$

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