If $3 x - 7 y = 10$ and $x y = - 1$ then value of $9 x^{2} + 49 y^{2}$ is?

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Solution

$9 x^{2} + 49 y^{2} = (3 x)^{2} + (7 y)^{2}$
$\Rightarrow (3 x + 7 y)^{2} - 2 \times 3 x \times 7 y$
or $(3 x + 7 y)^{2} - 2 \times 3 x \times 7 y = (3 x - 7 y)^{2} - 2 \times 3 x \times 7 y + 4 \times 3 x \times 7 y$
$\Rightarrow (3 x - 7 y)^{2} + 2 \times 3 x \times 7 y$
$= (3 x - 7 y)^{2} + 42 x y$
Given, $(3 x - 7 y) = 10$ & $x y = - 1$
$∴ 9 x^{2} + 49 y^{2} = 10^{2} + 42 \times (- 1) = 58$ .

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