Among the myriad of equations that exist, some, like the one at handβ x2+(y-3β2x)2=1 meaningβmay appear cryptic initially. However, these equations hold profound insights when dissected methodically. In this comprehensive article, we will delve deeply into the equation x2+(y-3β2x)2=1 meaning. Our mission is to elucidate its meaning, meticulously analyze its components, furnish you with a step-by-step solution, and underscore its relevance in mathematics and practical applications. So, fasten your seatbelts as we embark on a fascinating journey through the world of mathematics.

Before we delve into the intricate details of this equation, it is essential to recognize the critical role equations play in mathematics. They serve as powerful tools for expressing complex relationships between variables. The equation x^2 + (y β 3β2x)^2 = 1, despite its initial complexity, embodies a well-defined geometric shapeβspecifically, a circle. To truly grasp its significance, we must embark on a journey to break down its components, revealing the profound meaning concealed within.

## Breaking Down the Equation x2+(y-3β2x)2=1 Meaning

### Understanding the Terms

Our first step in comprehending the equation involves a meticulous analysis of its individual terms:

**x^2**: This term represents the square of the x-coordinateβa fundamental mathematical concept that underpins algebraic expressions.**(y β 3β2x)^2**: This portion introduces a layer of complexity. It incorporates two variables, x and y, and introduces the concept of a square root. In essence, it signifies the summation of the squares of the difference between y and three times the square root of two times x. This complex expression forms the crux of the equationβs geometric interpretation.

## Visualizing the Equation

To truly appreciate this equationβs essence, we must translate it into a visual representation. When we plot the points that satisfy this equation on a Cartesian plane, a remarkable pattern emergesβa perfect circle. This visual depiction serves as a crucial aid in comprehending the equationβs geometric significance.

## Finding the Center and Radius

Determining the center and radius of the circle described by this equation is pivotal in our exploration:

**Center**: The epicenter of this circle is positioned precisely at the origin, where both x and y are equal to 0. Consequently, the center is precisely located at the point (0, 0).**Radius**: As previously mentioned, the radius of this circle measures 1 unit. This vital measurement defines the distance from the center to any point along the circleβs circumference.

## Plotting the Circle

Now armed with knowledge about the center and radius, we can proceed to plot the circle on a Cartesian plane. The equation x^2 + (y β 3β2x)^2 = 1 encompasses all the points on the circleβs perimeter, providing a visual representation of its geometric characteristics.

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## Conclusion

In conclusion, the equation x2+(y-3β2x)2=1 meaning represents a circle within the Cartesian coordinate system. By dissecting its components, visualizing its geometric interpretation, and determining the center and radius, we have unveiled its hidden meaning and significance. Equations of this nature are foundational tools in mathematics and find applications in various domains, including geometry for precise shape descriptions and physics for modeling physical phenomena.