Question

If$(7a+8b)(7c-8d)=(7a-8b)(7c+8d)$ prove that $a:b=c:d$

Answer 1

prove that $a:b=c:d$
Given: $(7a+8b)(7c-8d)=(7a-8b)(7c+8d)$

If $\frac{a}{b}=\frac{c}{d}$ then $\frac{a+b}{a-b}=\frac{c+d}{c-d}$

[by componendo and dividendo]

$\Rightarrow \frac{7a+8b+7a-8b}{7a+8b-7a+8b}=\frac{7c+8d+7c-8d}{7c+8d-7c+8d}$

$\Rightarrow \frac{14a}{16b}=\frac{14c}{16d}$

$\Rightarrow \frac{a}{b}=\frac{c}{d}$
$\therefore a:b=d:d$

Hence, proved.