A) b = c
B) c = a
C) a = b
D) a + b + c = 0
E) (e) All the above
Correct Answer: E
Solution :
(e) The two lines will be identical if there exists some real number k such that \[{{b}^{3}}-{{c}^{3}}=k(b-c),\]\[{{c}^{3}}-{{a}^{3}}=k(c-a)\],\[{{a}^{3}}-{{b}^{3}}=k(a-b)\] Þ \[b-c=0\]or \[{{b}^{2}}+{{c}^{2}}+bc=k\] Þ \[c-a=0\]or \[{{c}^{2}}+{{a}^{2}}+ac=k\] Þ \[a-b=0\]or \[{{a}^{2}}+{{b}^{2}}+ab=k\] Þ \[b=c,c=a,a=b\]or \[{{b}^{2}}+{{c}^{2}}+bc={{c}^{2}}+{{a}^{2}}+ca\] Þ \[{{b}^{2}}-{{a}^{2}}=c(a-b)\Rightarrow b=a\]or \[a+b+c=0\].You need to login to perform this action.
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