Find x in terms of a b, and c: a/(x - a) + b/(x - b) = 2 c/(x

Q) Find X in terms of a b, and c: $\frac{a}{\times - a} + \frac{b}{\times - b} = \frac{2 c}{\times - c}$ , X ≠ a, b, c

Ans:

Given equation is:

$\frac{a}{(\times - a)} + \frac{b}{(\times - b)} = \frac{2 c }{(\times - c)}$

∴ $\frac{a (\times - b) + b (\times - a)}{(\times - a) (\times - b)} = \frac{2 c }{(\times - c)}$

∴ [a (X – b) + b (X – a)] (X – c) = 2 c (X – a) (X – b)

∴ (a X – a b + b X – a b )(X – c) = 2 c (X – a) (X – b)

∴ (a X + b X – 2 a b) (X – c) = 2 c (X² – a X – b X + a b)

∴ a X² + b X² – 2 a b X – a c X – b c X + 2 a b c = 2 c X² – 2 a c X – 2 b c X + 2 a b c

∴ a X² + b X² – 2 c X² – 2 a b X + a c X + b c X = 0

∴ X (a X + b X – 2 c X – 2 a b + a c + b c) = 0

∴ X [ X (a + b – 2 c ) – (2 a b – a c – b c )] = 0

∴ X = 0 and X (a + b – 2 c ) – (2 a b – a c – b c ) = 0

∴ X = 0 and X = $\frac{(2 a b - a c - b c )}{(a + b - 2 c)}$

Therefore, X = 0 and X = $\frac{(2 a b - a c - b c )}{(a + b - 2 c)}$ are the two roots of the given equation.

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