JEE Main & Advanced Mathematics Pair of Straight Lines Question Bank Bisectors of the angle between the lines, Point of intersection of the lines

  • question_answer If the bisectors of the angles between the pairs of lines given by the equation \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] and \[a{{x}^{2}}+2hxy+b{{y}^{2}}+\lambda ({{x}^{2}}+{{y}^{2}})=0\] be coincident, then \[\lambda =\]

    A)            a    

    B)            b

    C)            \[h\]                                        

    D)            Any real number

    Correct Answer: D

    Solution :

               Bisectors of \[a{{x}^{2}}+2hxy+b{{y}^{2}}=0\] are            \[\frac{{{x}^{2}}-{{y}^{2}}}{a-b}=\frac{xy}{h}\]                                                        .....(i)            and of \[a{{x}^{2}}+2hxy+b{{y}^{2}}+\lambda ({{x}^{2}}+{{y}^{2}})=0\]            i.e., \[(a+\lambda ){{x}^{2}}+2hxy+(b+\lambda ){{y}^{2}}=0\]are            \[\frac{{{x}^{2}}-{{y}^{2}}}{(a+\lambda )-(b+\lambda )}=\frac{xy}{h}\]                                        .....(ii)            Which is the same equation as equation (i). Hence for any \[\lambda \]belonging to real numbers, the lines will have same bisectors.

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