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smc

algebra senior

Problem

If

x y = a, x z = b,

and

y z = c

, and none of these quantities is

0

, then

x^{2} + y^{2} + z^{2}

equals

(A)

\frac{ab + a c + b c}{ab c}

(B)

\frac{a ^{2} + b ^{2} + c ^{2}}{ab c}

(C)

\frac{( a + b + c ) ^{2}}{ab c}

(D)

\frac{( ab + a c + b c ) ^{2}}{ab c}

(E)

\frac{( ab ) ^{2} + ( a c ) ^{2} + ( b c ) ^{2}}{ab c}

Solution

From the equations, we deduce

x = \frac{a}{y}, z = \frac{b}{x},

and

y = \frac{c}{z}

. Substituting these into the expression

x^{2} + y^{2} + z^{2}

thus gives

\frac{a ^{2}}{y ^{2}} + \frac{b ^{2}}{x ^{2}} + \frac{c ^{2}}{z ^{2}} = \frac{a ^{2} x ^{2} z ^{2} + b ^{2} y ^{2} z ^{2} + c ^{2} y ^{2} x ^{2}}{x ^{2} y ^{2} z ^{2}} = \frac{a ^{2} b ^{2} + b ^{2} c ^{2} + c ^{2} a ^{2}}{x ^{2} y ^{2} z ^{2}}

, so the answer is

\frac{( ab ) ^{2} + ( a c ) ^{2} + ( b c ) ^{2}}{ab c}

.

Final answer

E