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

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.