Circle Equations Diameter 12 Units Center On X-axis

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In this comprehensive guide, we delve into the fascinating world of circles and their equations. Specifically, we will explore the scenario where a circle has a diameter of 12 units, and its center is located on the x-axis. Our objective is to identify the equations that accurately represent such a circle. This exploration will involve understanding the standard form of a circle's equation, how the diameter relates to the radius, and how the center's location influences the equation. By the end of this discussion, you will have a solid grasp of how to determine the equation of a circle given its diameter and center's position.

To effectively tackle this problem, we must first understand the standard form of a circle's equation. The standard equation of a circle is given by:

(x - h)² + (y - k)² = r²

where:

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

This equation is derived from the Pythagorean theorem and describes the relationship between the coordinates of any point on the circle (x, y), the center of the circle (h, k), and the radius r. The distance between any point on the circle and the center is always equal to the radius. This foundational understanding is crucial for determining the correct equations for our specific circle.

In our case, we know that the circle's center lies on the x-axis. This means that the y-coordinate of the center (k) will be 0. Therefore, our equation simplifies to:

(x - h)² + y² = r²

This simplification is a key step in narrowing down the possible equations. We also know that the diameter of the circle is 12 units. The radius is half the diameter, so the radius (r) is 6 units. Therefore, r² is 36. Our equation now looks like this:

(x - h)² + y² = 36

The only remaining variable is h, the x-coordinate of the center. Different values of h will shift the circle horizontally along the x-axis, but the radius will remain constant at 6 units. To determine the correct equations, we need to identify the possible values of h based on the given options.

Now that we have established the basic form of the equation (x - h)² + y² = 36, we can analyze the provided list of equations. We need to look for equations that fit this form and have a radius squared (r²) equal to 36. The value of h will determine the horizontal position of the circle's center on the x-axis.

Let's consider a few scenarios:

  1. If the center of the circle is at the origin (0, 0), then h = 0, and the equation becomes:

    x² + y² = 36

    This is a valid equation for a circle with a diameter of 12 units centered at the origin.

  2. If the center of the circle is at (6, 0), then h = 6, and the equation becomes:

    (x - 6)² + y² = 36

    This represents a circle with a diameter of 12 units centered 6 units to the right of the origin.

  3. If the center of the circle is at (-6, 0), then h = -6, and the equation becomes:

    (x + 6)² + y² = 36

    This represents a circle with a diameter of 12 units centered 6 units to the left of the origin.

By substituting different values for h, we can generate a variety of equations that represent circles with a diameter of 12 units centered on the x-axis. The key is to ensure that the equation matches the form (x - h)² + y² = 36. When evaluating the list of equations, pay close attention to the sign of h, as it determines the direction of the horizontal shift.

To select the correct equations from the provided list, we need to compare each equation to the standard form we derived: (x - h)² + y² = 36. The crucial elements to look for are:

  1. The presence of squared terms: Both x and y should be squared.
  2. The coefficient of the squared terms: The coefficients of x² and y² should be 1.
  3. The constant term: The constant term on the right side of the equation should be 36 (since r² = 6² = 36).
  4. The value of h: The value of h determines the x-coordinate of the center. Look for terms like (x - h) or (x + h), where the sign will indicate the direction of the shift (positive h shifts the circle to the right, and negative h shifts it to the left).

For each equation in the list, carefully examine these elements. If an equation satisfies all these criteria, then it represents a circle with a diameter of 12 units centered on the x-axis. If an equation deviates from this form, it can be ruled out.

Let's consider some examples of how to analyze equations:

  • Example 1: x² + y² = 36

    This equation matches our standard form perfectly. Here, h = 0, so the center is at the origin (0, 0). The radius squared is 36, so the radius is 6. This is a valid equation.

  • Example 2: (x - 3)² + y² = 36

    This equation also matches our standard form. Here, h = 3, so the center is at (3, 0). The radius squared is 36, so the radius is 6. This is another valid equation.

  • Example 3: (x + 5)² + y² = 36

    This equation is in the correct form as well. Here, h = -5, so the center is at (-5, 0). The radius squared is 36, so the radius is 6. This is a valid equation.

  • Example 4: x² + y² = 144

    This equation has the correct form, but the constant term is 144, which means r² = 144 and r = 12. This represents a circle with a diameter of 24 units, not 12 units. Therefore, this is not a valid equation.

  • Example 5: (x - 2)² + (y + 2)² = 36

    This equation has a radius of 6, but the center is at (2, -2), not on the x-axis. The y-coordinate of the center is not 0, so this is not a valid equation.

By systematically analyzing each equation in this way, you can accurately identify the equations that represent a circle with a diameter of 12 units centered on the x-axis. Remember to focus on the standard form of the equation, the value of r², and the y-coordinate of the center.

In conclusion, determining the equations that represent a circle with a specific diameter and center location involves a clear understanding of the standard equation of a circle. By recognizing that the center lies on the x-axis, we can simplify the equation and focus on identifying the correct x-coordinate of the center (h). The radius, being half the diameter, provides the value for r², which is a constant in our case (36). By carefully comparing the provided equations to the standard form (x - h)² + y² = 36, we can accurately select the equations that fit the given criteria. Remember to pay close attention to the sign of h, the constant term, and the presence of squared terms with coefficients of 1. This methodical approach ensures that you can confidently identify the correct equations for circles with specific properties.

This exercise demonstrates the powerful connection between geometric properties and algebraic representations. By mastering the standard equation of a circle and its variations, you gain the ability to describe and analyze circles effectively in various mathematical contexts. The principles discussed here are fundamental to understanding more advanced concepts in geometry and analytic geometry.