Let $x,y$ are real numbers such that $x,x + 2y,2x + y$ form an A.P. while the numbers $(y + 1)^{2},x

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Let $x,y$ are real numbers such that $x,x + 2y,2x + y$ form an A.P. while the numbers $(y + 1)^{2},xy + 5,(x + 1)^{2}$ form a G.P., then $|x - y|$ is equal to

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Sol. [Key Concept: $x,x + 2y,2x + y$ in A.P. and the three squares in G.P. ⇒ solve the two conditions.]$$\begin{matrix} & & 2(x + 2y) & = x + (2x + y) \ & & & \Rightarrow x = 3y \ & & (y + 1)^{2}(x + 1)^{2} & = (xy + 5)^{2} \ & \Rightarrow & \Rightarrow & \Rightarrow (3y + 1)(y + 1) = \pm \left( 3y^{2} + 5 ight) \ & 	herefore & y & = 1	ext{~}	ext{or}	ext{~}3y^{2} + 2y + 3 = 0 \ & & y & = 1,x = 3 \end{matrix}$$

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