Suppose ^@ a \ne 0, b \ne 0, c \ne 0 ^@ and ^@ \dfrac{ a } { b } = \dfrac{ b } { c } = \dfrac{ c } { a } .^@ Find the value of ^@ \dfrac{ a - b + c } {a + b - c }. ^@
Answer:
^@ 1 ^@
- We need to find the value of ^@ \dfrac{ a - b + c }{ a + b - c } . ^@
Let ^@ \dfrac{ a } { b } = \dfrac{ b } { c } = \dfrac{ c } { a } = k ^@ - From ^@ \dfrac{ a } { b } = \dfrac{ b } { c } = \dfrac{ c } { a } = k, ^@ we get
^@ \begin{align} & a = bk, b = ck, \text{ and } c = ak \\ \implies & a = ak^3 \\ \implies & k^3 = 1 \\ \implies & k = 1 \\ \implies & a = b = c \end{align} ^@ - Now,
@^ \dfrac{ a - b + c } {a + b - c } = \dfrac{ a - a + a } { a + a - a } = 1 @^
