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Liquids in Liquids

Ifa:b=c:d, pr...

Question

If $a : b = c : d,$ prove that
$(9 a + 13 b) (9 c - 13 d)$
$= (9 c + 13 d) (9 a - 13 b)$

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Solution

Prove that $(9 a + 13 b) (9 c - 13 d)$ $= (9 c + 13 d) (9 a - 13 b)$

$∵ \frac{a}{b} = \frac{c}{d}$
Multiply both sides by $\frac{9}{13^{'}}$ we get
$\frac{9 a}{13 b} = \frac{9 c}{13 d}$

$\Rightarrow \frac{9 a + 13 b}{9 q - 13 b} = \frac{9 c + 13 d}{9 c - 13 d}$ [By componendo & Dividendo]

By doing cross multiplication, we get
$\Rightarrow (9 a + 13 b) (9 c - 13 d) = (9 a - 13 b) (9 c + 13 d)$

Hence,proved.

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