Have you ever encountered the enigmatic Equation x^2 + (y – 3√2x)^2 = 1 and pondered over its profound significance? In this article, we shall embark on a journey of exploration to unravel the meanings and implications of this enigmatic equation. Let us delve into its intricacies step by step and unlock the secrets it holds in the realm of mathematics.

Understanding The Meaning of x^2 + (y – 3√2x)^2 = 1

The equation x^2 + (y – 3√2x)^2 = 1 embodies a mathematical relationship between two variables, x and y. In simpler terms, it characterizes a distinct shape in a two-dimensional plane, famously known as an Ellipse.

An ellipse is a closed curve that closely resembles a stretched or compressed circle. The typical form of an ellipse’s equation is expressed as (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where the coordinates (h, k) represent the center of the ellipse, while a and b determine its size and shape.

Analyzing The Specific Equation

In our specific equation, the center of the ellipse lies at the point (0, 0), as no constants or variables directly influence the x and y terms. However, the term 3√2x presents a unique combination of the Square Root (√) and the product of 3 and the square root of 2 (√2), affecting the y-coordinate, leading it to fluctuate with varying values of x.

Unveiling The Solution

To solve the equation x^2 + (y – 3√2x)^2 = 1, let us walk through the steps to simplify and discover its solutions.

• Step 1: Expand the equation: x^2 + (y – 3√2x)^2 = 1 expands to x^2 + (y^2 – 6√2xy + 18x^2) = 1.
• Step 2: Combine like terms: (1 + 18)x^2 + y^2 – 6√2xy = 1 simplifies to 19x^2 + y^2 – 6√2xy = 1.
• Step 3: Rearrange the equation: 19x^2 – 6√2xy + y^2 = 1.
• Step 4: Factorize the equation: (3√2x – y)(3√2x – y) = 1.
• Step 5: Take the square root of both sides: 3√2x – y = ±1.
• Step 6: Solve for y: 3√2x = y ± 1.

Consequently, we obtain two separate lines in the x-y plane intersecting the ellipse defined by the original equation.

Intriguingly, the concept of an ellipse emerges as a fascinating subject in the realm of mathematics, and its presence in various fields remains astounding.

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• x^2 + (y – 3√2x)^2 = 1