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 One bisector of the angle between the lines given by \[a{{(x-1)}^{2}}+2h\,(x-1)y+b{{y}^{2}}=0\] is \[2x+y-2=0\]. The other bisector is

    A)            \[x-2y+1=0\]                        

    B)            \[2x+y-1=0\]

    C)            \[x+2y-1=0\]                        

    D)            \[x-2y-1=0\]

    Correct Answer: D

    Solution :

               We have \[a{{(x-1)}^{2}}+2h(x-1)y+b{{y}^{2}}=0\]            or \[a{{(x-1)}^{2}}+2h(x-1)(y-0)+b{{(y-0)}^{2}}=0\]            This equation represents a pair of straight lines intersecting at (1, 0). Therefore shifting the origin at    (1, 0), we have \[x=X+1,\,y=Y+0\] and the equation reduces to \[a{{X}^{2}}+2hXY+b{{Y}^{2}}=0\]                            .....(i)            One of the bisectors of the angles between the lines given by (i) is \[2x+y-2=0\]or \[2(X+1)+Y-2=0\] i.e. \[2X+Y=0\]. Since the bisector are always at right angle, therefore the other bisector is \[X-2Y=0\]            i.e., \[x-1-2y=0\]or \[x-2y-1=0\].

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