$I f 3 x + 5 y = 11 a n d x y = 2, find the value of 9 x^{2} + 25 y^{2}$

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Solution

$3 x + 5 y = 11; x y = 2 Squaring on both sides, (3 x + 5 y)^{2} = (11)^{2} \Rightarrow (3 x)^{2} + (5 y)^{2} + 2 \times 3 x \times 5 y = 121 \Rightarrow 9 x^{2} + 25 y^{2} + 30 x y = 121 \Rightarrow 9 x^{2} + 25 y^{2} + 30 \times 2 = 121 (∵ x y = 2) \Rightarrow 9 x^{2} + 25 y^{2} + 60 = 121 \Rightarrow 9 x^{2} + 25 y^{2} = 121 - 60 = 61 ∴ 9 x^{2} + 25 y^{2} = 61$

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