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Q.

If $\frac{xy}{x + y} = a, \frac{xz}{x + z} = b and \frac{yz}{y + z} = c,$

Where a, b, c all are non zero numbers, then x is equal to:

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a

\frac{2 abc}{ab + bc - ca}

b

\frac{2 abc}{ab + ac - bc}

c

\frac{2 abc}{ac + bc - ab}

d

\frac{2 abc}{ab + bc - ac}

answer is C.

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Detailed Solution

Concept: Finding the value of x , by using the given equations
Given:

\frac{xy}{x + y} = a, \frac{xz}{x + z} = b, \frac{yz}{y + z} = c

1)

\frac{xy}{x + y} = a

\frac{x + y}{xy} = \frac{1}{a}

\frac{1}{y} + \frac{1}{x} = \frac{1}{a}

2)

\frac{xz}{x + z} = b

\frac{(x + z)}{xz} = b

3)

\frac{yz}{y + z} = c

\frac{y + z}{yz} = \frac{1}{c}

\frac{1}{z} + \frac{1}{y} = \frac{1}{c}

Adding the equations,

\frac{1}{y} + \frac{1}{x} + \frac{1}{z} + \frac{1}{x} = \frac{1}{a} + \frac{1}{b}

(\frac{1}{z} + \frac{1}{y}) + \frac{2}{x} = \frac{1}{a} + \frac{1}{b}

\frac{1}{c} + \frac{2}{x} = \frac{1}{a} + \frac{1}{b}

\frac{2}{x} = \frac{1}{a} + \frac{1}{b} - \frac{1}{c}

\frac{2}{x} = \frac{bc + ac - ab}{abc}

\frac{x}{2} = \frac{abc}{bc + ac - ab}

x = \frac{2 abc}{bc + ac - ab}

Hence, the correct option is (3).

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