Question : If $\frac{x}{y}=\frac{a+2}{a-2}$, then the value of $\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ is:
Option 1: $\frac{4a}{a^{2}+2}$
Option 2: $\frac{2a}{a^{2}+2}$
Option 3: $\frac{4a}{a^{2}+4}$
Option 4: $\frac{2a}{a^{2}+4}$
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Correct Answer: $\frac{4a}{a^{2}+4}$
Solution :
Given: $\frac{x}{y}=\frac{a+2}{a-2}$
Let $x = a+2$ and $y=a-2$,
Now, $\frac{x^{2}-y^{2}}{x^{2}+y^{2}}$ = $\frac{(a+2)^{2}-(a-2)^{2}}{(a+2)^{2}+(a-2)^{2}}$
= $\frac{(a^{2}+4a+4)-(a^{2}-4a+4)}{(a^{2}+4a+4)+(a^{2}-4a+4)}$
= $\frac{8a}{2a^{2}+8}$
= $\frac{4a}{a^{2}+4}$
Hence, the correct answer is $\frac{4a}{a^{2}+4}$.
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