Key Concepts and Formulas

• Pair of Straight Lines: The equation $Ax^2 + Bxy + Cy^2 = 0$ represents a pair of straight lines passing through the origin.
• Angle Bisectors of $xy=0$: The angle bisectors of the lines $x=0$ and $y=0$ are $y=x$ and $y=-x$.
• Line belonging to Pair: If a line $y=mx$ is part of the pair of lines represented by $Ax^2 + Bxy + Cy^2 = 0$, then substituting $y=mx$ into the equation must result in an identity (i.e., the equation becomes true for all values of $x$).

Step-by-Step Solution

Step 1: Identify the angle bisectors of the lines $xy=0$.

The equation $xy=0$ represents the pair of lines $x=0$ and $y=0$, which are the coordinate axes. The angle bisectors of the coordinate axes are $y=x$ and $y=-x$. This is a standard result.

Step 2: State the given equation and the condition.

The given equation representing a pair of straight lines is $m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0$. We are given that one of the lines represented by this equation is either $y=x$ or $y=-x$.

Step 3: Substitute $y=x$ into the given equation.

If $y=x$ is one of the lines, substituting $y=x$ into the given equation must make the equation an identity: $m(x)^2 + (1-m^2)x(x) - m(x)^2 = 0$ $mx^2 + (1-m^2)x^2 - mx^2 = 0$ $(m + 1 - m^2 - m)x^2 = 0$ $(1-m^2)x^2 = 0$

Step 4: Solve for $m$ when $y=x$ is a solution.

For $(1-m^2)x^2 = 0$ to be true for all $x$, we must have $1-m^2 = 0$. Thus, $m^2 = 1$ $m = \pm 1$

Step 5: Substitute $y=-x$ into the given equation.

If $y=-x$ is one of the lines, substituting $y=-x$ into the given equation must make the equation an identity: $m(-x)^2 + (1-m^2)x(-x) - m(x)^2 = 0$ $mx^2 - (1-m^2)x^2 - mx^2 = 0$ $(m - 1 + m^2 - m)x^2 = 0$ $(m^2 - 1)x^2 = 0$

Step 6: Solve for $m$ when $y=-x$ is a solution.

For $(m^2 - 1)x^2 = 0$ to be true for all $x$, we must have $m^2 - 1 = 0$. Thus, $m^2 = 1$ $m = \pm 1$

Step 7: Determine the correct value of $m$ from the options.

In both cases, we get $m = \pm 1$. From the given options, $m=1$ is one of the choices.

Common Mistakes & Tips

• Remember the Identity: The most common mistake is to solve for $x$ after substituting $y=x$ or $y=-x$. The key is that the equation must be identically zero for all $x$, meaning the coefficient of $x^2$ must be zero.
• Sign Errors: Be careful with signs when substituting $y=-x$.
• Consider Both Bisectors: Although in this case both bisectors give the same values for $m$, always check both $y=x$ and $y=-x$.

Summary

We found the angle bisectors of the lines $xy=0$ to be $y=x$ and $y=-x$. By substituting these into the given equation $m{y^2} + \left( {1 - {m^2}} \right)xy - m{x^2} = 0$, we found that $m = \pm 1$. The option $m=1$ is available, so that is the correct answer.

The final answer is $\boxed{1}$, which corresponds to option (A).