Have you ever come across the equation x^2 + (y – 3√2x)^2 = 1 and wondered what it signifies? In this article, we will delve into the meaning and implications of this equation. We will break it down step by step, exploring its components and unraveling its significance in the realm of mathematics. So, let’s embark on this journey of discovery and understanding.
The equation x^2 + (y – 3√2x)^2 = 1 represents a mathematical relationship between the variables x and y. In English, this equation describes a specific shape in a two-dimensional plane. This shape is called an ellipse.
An ellipse is a closed curve that resembles a stretched or squashed circle. The equation of an ellipse is typically written in the form (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h, k) represents the coordinates of the center of the ellipse, and a and b determine the size and shape of the ellipse.
In this specific equation, the center of the ellipse is at the point (0, 0) since there are no constants or variables directly affecting the x and y terms. The term 3√2x is a combination of the square root (√) and the number 3 multiplied by the square root of 2 (√2). This term affects the y-coordinate, causing it to change as x varies.
By substituting the values, you can determine the length and orientation of the major and minor axes of the ellipse, which determine its shape.
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To solve the equation x^2 + (y – 3√2x)^2 = 1, we can work through the steps to simplify and find the solutions.
- Expand the equation: x^2 + (y – 3√2x)^2 = 1 x^2 + (y^2 – 6√2xy + 18x^2) = 1
- Combine like terms: (1 + 18)x^2 + y^2 – 6√2xy = 1 19x^2 + y^2 – 6√2xy = 1
- Rearrange the equation: 19x^2 – 6√2xy + y^2 = 1
- Factorize: (3√2x – y)(3√2x – y) = 1
- Take the square root of both sides: 3√2x – y = ±1
- Solve for y: 3√2x = y ± 1
Therefore, the solutions for y are: y = 3√2x + 1 y = 3√2x – 1
These equations represent two separate lines in the x-y plane that intersect the ellipse defined by the original equation.