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JEE Main & Advanced Mathematics Straight Line Question Bank Slope of line, Equation of line in different forms

  • question_answer
    The equations \[(b-c)x+(c-a)y+(a-b)=0\] and \[({{b}^{3}}-{{c}^{3}})x+({{c}^{3}}-{{a}^{3}})y+{{a}^{3}}-{{b}^{3}}=0\] will represent the same line, if

    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 \[a+b+c=0\].


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