Arithmetic Mean of Square Root Fractions

CAT 2023 Slot 1 · QA · Hard · Algebra

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For some positive and distinct real numbers x,yx, y and zz, if 1y+z\frac{1}{\sqrt{y}+\sqrt{z}} is the arithmetic mean of 1x+z\frac{1}{\sqrt{x}+\sqrt{z}} and 1x+y\frac{1}{\sqrt{x}+\sqrt{y}}, then the relationship which will always hold true, is

  1. A.

    x,y\sqrt{x}, \sqrt{y} and z\sqrt{z} are in arithmetic progression

  2. B.

    x,z\sqrt{x}, \sqrt{z} and y\sqrt{y} are in arithmetic progression

  3. C.

    y,xy, x and zz are in arithmetic progression

  4. D.

    x,yx, y and zz are in arithmetic progression

Answer

C

Explanation

Given: 2y+z=1x+z+1x+y\frac{2}{\sqrt{y}+\sqrt{z}} = \frac{1}{\sqrt{x}+\sqrt{z}} + \frac{1}{\sqrt{x}+\sqrt{y}} 2y+z=2x+y+zx+xy+xz+yz\frac{2}{\sqrt{y}+\sqrt{z}} = \frac{2\sqrt{x} + \sqrt{y} + \sqrt{z}}{x + \sqrt{x}\sqrt{y} + \sqrt{x}\sqrt{z} + \sqrt{y}\sqrt{z}}

Cross-multiplying: 2(x+xy+xz+yz)=(y+z)(2x+y+z)2(x + \sqrt{x}\sqrt{y} + \sqrt{x}\sqrt{z} + \sqrt{y}\sqrt{z}) = (\sqrt{y}+\sqrt{z})(2\sqrt{x} + \sqrt{y} + \sqrt{z}) 2x+2xy+2xz+2yz=2xy+y+yz+2xz+yz+z2x + 2\sqrt{x}\sqrt{y} + 2\sqrt{x}\sqrt{z} + 2\sqrt{y}\sqrt{z} = 2\sqrt{x}\sqrt{y} + y + \sqrt{y}\sqrt{z} + 2\sqrt{x}\sqrt{z} + \sqrt{y}\sqrt{z} + z 2x+2yz=y+z+2yz2x + 2\sqrt{y}\sqrt{z} = y + z + 2\sqrt{y}\sqrt{z} 2x=y+z2x = y + z

Thus, y,x,zy, x, z are in Arithmetic Progression.

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