If (7a + 8b) (7c – 8d) = (7a – 8b) (7c + 8d);
Prove that a: b = c: d

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Solution

The given can the rewritten as
$\frac{7 a + 8 b}{7 a} = \frac{7 c + 8 d}{7 c - 8 d}$
Applying componendo and dividendo, we have
$\frac{7 a + 8 b + 7 a - 8 b}{7 a + 8 b - 7 a + 8 b} = \frac{7 c + 8 d + 7 c - 8 d}{7 c + 8 d - 7 c + 8 d}$
$\frac{14 a}{16 b} = \frac{14 c}{16 d}$
$\frac{a}{b} = \frac{c}{d}$

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