A) \[\frac{x-c}{(x-a)(x-b)}\]
B) \[\frac{x-a}{(x-b)(x-c)}\]
C) \[\frac{(x-c)}{(x-b)(a-x)}\]
D) None of the above
Correct Answer: A
Solution :
\[\frac{a-c}{(a-b)(x-a)}+\frac{b-c}{(b-a)(x-b)}\] \[=\frac{(a-c)(x-b)-(b-c)(x-a)}{(a-b)(x-a)(x-b)}\] \[=\frac{ax-ab-xc+bc-(bx-ab-cx+ac)}{(a-b)(x-a)(x-b)}\] \[=\frac{ax+bc-bx-ac}{(a-b)(x-a)(x-b)}\] \[=\frac{x(a-b)+c(b-a)}{(a-b)(x-a)(x-b)}\] \[=\frac{(x-c)(a-b)}{(a-b)(x-a)(x-b)}=\frac{(x-c)}{(x-a)(x-b)}\]
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