If \(\frac{2a^2-3b^2}{a^2+b^2}=\frac{2}{41}\), then \(a:b\) =
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 5 : 4
Cross-multiplying turns the given fraction into an equation in \(a^{2}\) and \(b^{2}\), from which the ratio \(a^{2}:b^{2}\), and then \(a:b\), follows.
\(41(2a^{2}-3b^{2})=2(a^{2}+b^{2})\)
\(82a^{2}-123b^{2}=2a^{2}+2b^{2}\)
\(80a^{2}=125b^{2}\)
Ratio of the squares: \(\displaystyle\frac{a^{2}}{b^{2}}=\frac{125}{80}=\frac{25}{16}\)
Square roots of both terms: \(\displaystyle\frac{a}{b}=\frac{5}{4}\), so \(a:b=5:4\)
Tip: The working first gives \(a^{2}:b^{2}=25:16\), which is Option (2); the question asks for \(a:b\), so take the square root of both terms.