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If 5x+6y5u+6v...

Question

If $\frac{5 x + 6 y}{5 u + 6 v} = \frac{5 x - 6 y}{5 u - 6 v};$
then prove that $x : y = u : v$

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Solution

Concept of componendo and dividendo to be used
Given: $\frac{5 x + 6 y}{5 u + 6 v} = \frac{5 x - 6 y}{5 u - 6 v}$
When you replace the denominator of the first ratio with the numerator of the second ratio,the two ratio remains proportional to each other,
If $a : b = c : d,$ then $a : c = b : d$
So, $\frac{5 x + 6 y}{5 u + 6 v} = \frac{5 x - 6 v}{5 u - 6 v}$
$\Rightarrow \frac{5 x + 6 y + 5 x - 6 y}{5 x + 6 y - 5 x + 6 y} = \frac{5 u + 6 v + 5 u - 6 v}{5 u + 6 v - 5 u + 6 v}$
$\Rightarrow \frac{10 x}{12 y} = \frac{10 u}{12 v}$
$\Rightarrow \frac{x}{y} = \frac{u}{v}$
$\Rightarrow x : y = u : v$
Hence,proved.

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