Question

If one of the lines of \[m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0\] is bisector of the angle between the lines \[xy = 0\], then m is
A.1
B.2
C.\[\dfrac{{ - 1}}{2}\]
D.-1

Answer 1

Hint: Given line is a pair of straight lines. And one of its lines is a bisector of angle between the lines \[xy = 0\]. Thus we will compare the pair of straight lines with the respective lines and then will find the value of m.

Complete step-by-step answer:
Line \[xy = 0\] means bisector of coordinate system.

Thus, x=y or x=-y are the two lines.
Now given that,
\[m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0\]
Multiplying xy with the middle terms,
\[
   \Rightarrow m{y^2} + xy - {m^2}xy - m{x^2} = 0 \\
   \Rightarrow y(my + x) - mx(my + x) = 0 \\
   \Rightarrow \left( {y - mx} \right)\left( {my + x} \right) = 0 \\
\]
Thus , two lines that appear are
\[ \Rightarrow y - mx = 0\] or \[my + x = 0\]
\[ \Rightarrow y = mx\] or \[y = - \dfrac{x}{m}\]
Thus comparing with the two lines above \[m = \pm 1\].
Thus correct options are A and D.

Note: In this problem the key point is only that the line \[xy = 0\] is the coordinate system and students should know the two lines of that system. And need to compare the pair of straight lines with it to get the value of m.