Circle Equation X² + Y² - 2x - 8 = 0 Analysis

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In this article, we will delve into the fascinating world of circles and explore the properties of a specific circle defined by the equation x² + y² - 2x - 8 = 0. We will analyze this equation to determine key characteristics such as its radius, center, and its position relative to the coordinate axes. By understanding these features, we can gain a deeper appreciation for the geometry of circles and their representation in the Cartesian plane.

Decoding the Circle Equation

To begin our exploration, let's first understand the general equation of a circle. The standard form of a circle's equation is given by:

(x - h)² + (y - k)² = r²

Where:

  • (h, k) represents the coordinates of the circle's center.
  • r represents the radius of the circle.

Our given equation is x² + y² - 2x - 8 = 0. To extract meaningful information, we need to transform this equation into the standard form. This involves a process called completing the square.

Completing the Square

Completing the square is a powerful algebraic technique used to rewrite quadratic expressions in a more convenient form. In our case, it will help us reveal the center and radius of the circle. Let's apply this technique to our equation:

  1. Group the x-terms and y-terms:

    (x² - 2x) + y² = 8

  2. Complete the square for the x-terms:

    To complete the square for x² - 2x, we need to add and subtract the square of half the coefficient of the x-term. The coefficient of our x-term is -2, so half of it is -1, and the square of -1 is 1. Therefore, we add and subtract 1:

    (x² - 2x + 1 - 1) + y² = 8

  3. Rewrite the x-terms as a squared expression:

    The expression x² - 2x + 1 is a perfect square trinomial and can be rewritten as (x - 1)²:

    ((x - 1)²) - 1 + y² = 8

  4. Isolate the squared terms and the constant term:

    Move the -1 to the right side of the equation:

    (x - 1)² + y² = 9

Now our equation is in the standard form:

(x - 1)² + (y - 0)² = 3²

Unveiling the Circle's Properties

By comparing our transformed equation with the standard form, we can readily identify the circle's center and radius. The equation (x - 1)² + (y - 0)² = 3² tells us a great deal about this circle. The center of the circle is (h, k) = (1, 0), and the radius of the circle is r = 3. This equation is the key to understanding the circle's characteristics. We can now answer some fundamental questions about this circle. With this information, we can now evaluate the given statements.

Analyzing the Statements

Let's examine each statement in light of our findings:

  1. The radius of the circle is 3 units.

    This statement is TRUE. We directly obtained the radius as r = 3 from the standard form of the equation.

  2. The center of the circle lies on the x-axis.

    This statement is also TRUE. The center of the circle is (1, 0). Any point with a y-coordinate of 0 lies on the x-axis. Therefore, the circle's center lies on the x-axis.

  3. The center of the circle lies on the y-axis.

    This statement is FALSE. The center of the circle is (1, 0). Any point with an x-coordinate of 0 lies on the y-axis. Since the x-coordinate of the center is 1, it does not lie on the y-axis.

  4. The standard form...

    This incomplete statement requires a bit more context. To discuss the standard form fully, we should restate it as, The standard form of the equation of the circle is (x - 1)² + y² = 9. This statement is TRUE, as we derived this form through completing the square. This is a critical understanding for anyone delving into analytic geometry and conic sections.

Visualizing the Circle

To further solidify our understanding, it's helpful to visualize the circle. Imagine a circle in the Cartesian plane with its center at the point (1, 0) and a radius of 3 units. This circle will intersect the x-axis at two points and extend 3 units above and below the x-axis. Understanding the geometric representation of the equation reinforces our algebraic analysis.

Key Takeaways

From our analysis, we've learned the following key properties of the circle x² + y² - 2x - 8 = 0:

  • The circle has a radius of 3 units.
  • The center of the circle is located at the point (1, 0).
  • The center of the circle lies on the x-axis.
  • The standard form of the circle's equation is (x - 1)² + y² = 9.

These properties provide a complete description of the circle's position and size in the coordinate plane. The process of completing the square was instrumental in revealing these properties. This technique is a cornerstone of many algebraic manipulations.

Conclusion

By carefully analyzing the equation x² + y² - 2x - 8 = 0, we have successfully determined its key properties: its radius, center, and position relative to the axes. We achieved this by transforming the equation into its standard form using the technique of completing the square. This exercise demonstrates the power of algebraic manipulation in unveiling geometric information. Understanding the interplay between algebra and geometry is crucial for success in mathematics and related fields. This exploration highlights the beauty and precision of mathematical analysis, enabling us to decipher the hidden characteristics of geometric shapes from their equations.