In this article, we will explore the process of determining the area of the region inside the curve defined by the equation x2xyy21. This problem involves the manipulation of a complex equation to express it in a more manageable form. We will utilize transformations and parametric equations to solve the problem and ultimately find the area of the ellipse.

1. Initial Transformation: Cubic Root of Unity

We start by noting that the given equation x2-xyy2xjy2,

where v is the cubic root of unity, can be rewritten as:

xjy1

Euler’s formula states that xjyeit, where t is a parameter.

2. Parametric Representation of the Curve

We can rewrite the rewritten equation as:

xjycos t and isin t.

This gives us a parametric representation of our curve:

xjycos t and isin t.

To further simplify, we have:

ilfrac{1}{sqrt{3}}12j

Therefore, we get:

xjycos t and isin tcos t dfrac{1}{sqrt{3}}12jsin t.

Solving for x and y as functions of the parameter t, we get:

3. Calculating the Area

To find the area, we recognize y as a function of x for t u2208 [0, u03C0]. The area can be double the integral Area2I of that function over [0, u03C0]. We can write:

Thus, the requested area is:

dfrac{2pi}{sqrt{3}}

4. Alternative Method Using Rotation and Transformations

Alternatively, if the region is an ellipse rotated 90 degrees and centered at the origin, the major and minor axes will intersect with yx and y-x. Assuming a tilt of 45 degrees from the origin, the major axis is at 11 and the minor axis at -√√. The area of the ellipse can be found as:

We can rotate the coordinate frame by an angle of frac{u03C0}{4} using the 2D rotation matrix:

The resulting equation simplifies to:

3u^2v^22.

From this, we find the semimajor and semiminor axis lengths. When v0, upm sqrt{frac{2}{3}}. When u0, vpm sqrt{2}. These lengths are the distances from the origin to the axis intercepts.

The area of an ellipse is given by the product of these lengths multiplied by u03C0.(The equation for the area is: Asqrt{frac{2}{3}}sqrt{2}u03C0frac{2u03C0}{sqrt{3}}).

5. Conclusion

By leveraging transformations and parametric equations, we were able to determine the area of the ellipse defined by the equation x^2xyy^21. Whether through the initial transformation or alternative methods involving rotation, we found that the area is:

frac{2u03C0}{sqrt{3}}