If $x_{1}, x_{2}$ are the roots of $x^{2} - x + K = 0$ and $x_{3}, x_{4}$ are the roots of $x^{2} - 4 x + L = 0$ such that $x_{1}, x_{2}, x_{3}, x_{4}$ are in G.P. Then the product of the integral values of K and L is ______

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answer is 64.

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Detailed Solution

Let the common ratio of G.P. be ‘ r ’ then $x_{2} = x_{1} r, x_{3} = x_{1} r^{2}, x_{4} = x_{1} r^{3}$ and
$x_{1} + x_{2} = 1, x_{1} x_{2} = K, x_{3} + x_{4} = 4, x_{3} . x_{4} = L$

$x_{1} + x_{1} r = 1, x_{1} r^{2} + x_{1} r^{3} = 4,$

$x_{1} (1 + r) = 1, x_{1} r^{2} (1 + r) = 4,$

$∴ r = - 2$ and $x_{1} = - 1$
$\Rightarrow K = - 2, L = - 32$

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