If a + b + c = 0, then find the value of
[(b–c)/a + (c–a)/b + (a–b)/c].[a/(b–c) + b/(c–a) + c/(a–b)]
(In reply to
re: Solution by Richard)
My opinion is in conformity with the observations made by Richard. I do not believe that it would be feasible for me to match Bractal's formatting. <o:p></o:p>
An alternate solution to the puzzle is provided below:<o:p> </o:p>
- Denoting the given expression as S*T; where S = (b � c)/a + (c � a)/b + (a � b)/c; and T = a/(b � c) + b/(c � a) + c/(a � b); we obtain S = -(a-b)(b-c)(c-a)/(abc), upon simplification.<o:p></o:p>
- Substituting, p = a-b; q = b-c and r = c-a, we observe that p-q = a-2b+c = -3b ( since a+b+c =0).
Similarly, q-r = -3c and r-p = -3a, so that: - T = - (1/3)*[(r-p)/ q +(p-q)/r + (q-r)/p]= -(1/3)[(p-q)(q-r)(r-p)/(pqr)]
- = -(1/3)[(-3b)(-3c)(-3a)][1/(a-b)(b-c)(c-a)]
- = (9abc)/ [(a-b)(b-c)(c-a)]
= 9/S; so that ST = 9, which is the required value of the given expression.
<o:p> </o:p>
Edited on February 4, 2024, 12:37 am
re(2): Solution_ An Alternate Solution To The Puzzle
