Question : The third proportional of the following numbers $(x-y)^2, (x^2-y^2)^2$ is:
Option 1: $(x+y)^3(x-y)^2$
Option 2: $(x+y)^4(x-y)^2$
Option 3: $(x+y)^2(x-y)^2$
Option 4: $(x+y)^2(x-y)^3$
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Correct Answer: $(x+y)^4(x-y)^2$
Solution :
Given:
Two numbers are $(x-y)^2, (x^2-y^2)^2$.
So, let's take $a=(x-y)^2$ = 1st proportion,
$b=(x^2-y^2)^2$ = 2nd proportion
Now, $c$ = third proportion = $\frac{b^2}{a}=\frac{((x^2-y^2)^2)^2}{(x-y)^2}=\frac{(x-y)^4 (x+y)^4}{(x-y)^2}=(x+y)^4(x-y)^2$
Hence, the correct answer is $(x+y)^4(x-y)^2$.
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