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Implicit Differentiation

Let x and y b...

Question

Let x and y be real numbers such that $x, x + 2 y, 2 x + y$ form an A.P. while the numbers $(y + 1)^{2}, x y + 5, (x + 1)^{2}$ form a G.P. Find $x + y$ ?

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Solution

The condition for A.P. $2 (x + 2 y) = 3 x + y \Rightarrow x = 3 y$ (1)
The condition for G.P. $(x y + 5)^{2} = (x + 1)^{2} (y + 1)^{2} \Rightarrow (3 y^{2} + 5)^{2} = [(3 y + 1) (y + 1)]^{2} = 0, b y (1)$
$\Rightarrow (y - 1) (3 y^{2} + 2 y + 3) = 0, ∴ y = 1, - 1 + 2 \sqrt{2 i}, - 1 - 2 \sqrt{2} i$
But given $x, y$ are real $∴ y = 1$ and $x = 3 ∴ x + y = 4$

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