Question
If (7a + 8b) (7c – 8d) = (7a – 8b) (7c + 8d), prove that a: b = c: d.
✓
Answer
Given $\frac{7 a+8 b}{7 a-8 b}=\frac{7 c+8 d}{7 c-8 d}$ ...(Invertendo)
Applying componendo and dividendo
$\frac{7 a+8 b+7 a-8 b}{7 a+8 b-7 a+8 b}=\frac{7 c+8 d+7 c-8 d}{7 c+8 d-7 c+8 d}$
$\Rightarrow \frac{14 a}{16 b}=\frac{14 c}{16 d}$
$\Rightarrow \frac{a}{b}=\frac{c}{d}$
Hence a : b = c : d
Applying componendo and dividendo
$\frac{7 a+8 b+7 a-8 b}{7 a+8 b-7 a+8 b}=\frac{7 c+8 d+7 c-8 d}{7 c+8 d-7 c+8 d}$
$\Rightarrow \frac{14 a}{16 b}=\frac{14 c}{16 d}$
$\Rightarrow \frac{a}{b}=\frac{c}{d}$
Hence a : b = c : d
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