Introduction
When the axes are bisectors of the angles between them, the equation of the curve becomes simpler. In this case, the equation 7x^2 4xy 4y^2=0 can be simplified using this property.

Rotation of Axes
To simplify the equation, we need to rotate the axes. Let the new axes be X and Y, and let the angle between the original and new X-axis be theta. Then we have:

X = x cos(theta) + y sin(theta)
Y = -x sin(theta) + y cos(theta)

Transformation of Equation
Using these equations, we can transform the original equation into an equation in terms of X and Y. Substituting x and y with their respective expressions in terms of X and Y, we get:

7(X cos(theta) + Y sin(theta))^2 + 4(X cos(theta) + Y sin(theta))(-X sin(theta) + Y cos(theta)) + 4(-X sin(theta) + Y cos(theta))^2 = 0

Simplifying this equation, we get:

7X^2 - 6XY + 7Y^2 = 0

Conclusion
Thus, the equation 7x^2 4xy 4y^2=0 becomes 7X^2 - 6XY + 7Y^2 = 0 when the axes are bisectors of the angles between them. This new equation is simpler than the original equation and can be easily studied and analyzed.