If \((x+y): (x-y)=7: 3\), then find the ratio of \((x^2+ y^2): (x^2-y^2)\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 29 : 21
A ratio fixes the two quantities up to a common multiple \(k\). Writing \(x+y\) and \(x-y\) with \(k\) gives \(x\) and \(y\) themselves, and \(k\) then cancels in the ratio asked for.
Finding \(x\) and \(y\):
\(x+y=7k\) (1)
\(x-y=3k\) (2)
Adding (1) and (2), the \(y\) terms cancel: \(2x=10k\), so \(x=5k\)
Subtracting (2) from (1): \(2y=4k\), so \(y=2k\)
The ratio asked for:
\(x^{2}+y^{2}=25k^{2}+4k^{2}=29k^{2}\)
\(x^{2}-y^{2}=25k^{2}-4k^{2}=21k^{2}\)
\((x^{2}+y^{2}):(x^{2}-y^{2})=29:21\)