Question : If $ \frac{(5x\:-\:y)}{(5x\:+\:y)}=\frac{3}{7},$ what is the value of $\frac{(4x^{2}\:+\:y^{2}\:–\:4xy)}{(9x^{2}\:+\:16y^{2}\:+\:24xy)}$?
Option 1: $0$
Option 2: $\frac{3}{7}$
Option 3: $\frac{18}{49}$
Option 4: $\frac{1}{6}$
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Correct Answer: $0$
Solution :
Given: $\frac{(5x\:-\:y)}{(5x\:+\:y)}=\frac{3}{7}$
$⇒35x-7y=15x+3y$
$⇒20x-10y=0$
$⇒2x-y=0$---------------(i)
$\frac{(4x^{2}\:+\:y^{2}\:-\:4xy)}{(9x^{2}\:+\:16y^{2}\:+\:24xy)}$
= $\frac{(2x\:-\:y)^{2}}{(9x^{2}\:+\:16y^{2}\:+\:24xy)}$
Since from equation (i), we know $2x-y=0$, substituting the value in the above equation, we get,
$=\frac{0}{(9x^{2}\:+\:16y^{2}\:+\:24xy)}=$ 0
Hence, the correct answer is $0$.
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