If (2a^2 - 3b^2)/(a^2 + b^2) = 2/41, then a : b = Option 1: 5 : 4 Option 2: 25 : 16 Option 3: 16 : 25 Option 4: 4 : 5

Answer: 5 : 4. Solution: Cross multiply: 41(2a^2 - 3b^2) = 2(a^2 + b^2). 82a^2 - 123b^2 = 2a^2 + 2b^2, so 80a^2 = 125b^2. a^2/b^2 = 125/80 = 25/16, so a/b = 5/4. Therefore a : b = 5 : 4.

Arithmetic

Ratio, Proportion & Variation

Medium

If $\frac{2a^2 - 3b^2}{a^2 + b^2} = \frac{2}{41}$ , then $a : b =$

5 : 4

25 : 16

16 : 25

4 : 5

✅ Correct Option: 1

Cross multiply: $41(2a^2 - 3b^2) = 2(a^2 + b^2)$ .

$82a^2 - 123b^2 = 2a^2 + 2b^2$ , so $80a^2 = 125b^2$ .

$\frac{a^2}{b^2} = \frac{125}{80} = \frac{25}{16}$ , so $\frac{a}{b} = \frac{5}{4}$ .

Therefore $a : b = 5 : 4$ .

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If $\frac{2a^2 - 3b^2}{a^2 + b^2} = \frac{2}{41}$ , then $a : b =$