Question

If $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$ prove that $$\left(\frac{a^2}{b^2} + \frac{c^2}{d^2} + \frac{e^2}{f^2}\right) = \left(\frac{ac}{bd} + \frac{ce}{df} + \frac{ae}{bf}\right)$$

Answer 1

Given: $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f}$$

To prove: $$\left(\frac{a^2}{b^2} + \frac{c^2}{d^2} + \frac{e^2}{f^2}\right) = \left(\frac{ac}{bd} + \frac{ce}{df} + \frac{ae}{bf}\right)$$

Proof:

1. Let $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = k$$.
2. $$\text{L.H.S.} = \left(\frac{a}{b}\right)^2 + \left(\frac{c}{d}\right)^2 + \left(\frac{e}{f}\right)^2 = k^2 + k^2 + k^2 = 3k^2$$
3. $$\text{R.H.S.} = \left(\frac{a}{b}\right)\left(\frac{c}{d}\right) + \left(\frac{c}{d}\right)\left(\frac{e}{f}\right) + \left(\frac{a}{b}\right)\left(\frac{e}{f}\right) = k \cdot k + k \cdot k + k \cdot k = 3k^2$$
4. $$\text{L.H.S.} = \text{R.H.S.}$$

Hence proved.