Animated Solution for Mathematics - Straight Lines: If one of the lines of $m{y}^2+(1-m^2)xy-m{x}^2=0$ is a bisector of the angle between the lines $xy=0$, then $m$ is

The given equation is $m{y}^2+(1-m^2)xy-m{x}^2=0$. This represents a pair of straight lines passing through the origin.

The problem references the lines $xy=0$. By the zero-product property, this equation represents the union of the x-axis ($y=0$) and the y-axis ($x=0$).

These axes intersect at the origin at an angle of $90^{\circ }$. The angle bisectors of these axes are the lines $y=x$ and $y=-x$. These lines divide the coordinate plane into $45^{\circ }$ and $135^{\circ }$ angles.

We are given that one of the lines represented by the quadratic equation is a bisector of the axes. This implies that the line $y=x$ (or $y=-x$) must satisfy the equation $m{y}^2+(1-m^2)xy-m{x}^2=0$.

Substituting $y=x$ into the equation, we obtain:

Expanding and simplifying the terms, we get:

The terms $m{x}^2$ and $-m{x}^2$ cancel each other out perfectly. This leaves us with the following expression:

For this equation to represent the entire line $y=x$, the condition must hold for all values of $x$. Therefore, the coefficient of $x^2$ must be zero.

Setting the coefficient to zero, we find:

Thus, the possible values for $m$ are $1$ or $-1$. By utilizing the geometric property of the angle bisectors, we have successfully reduced a complex algebraic form to a simple constraint.