- Determine that a circle with diameter 12 has a radius of 6, so its equation in standard form must have \(r^2 = 36\).
- Recognize that a center on the \(x\)-axis implies the equation is of the form \((x - h)^2 + y^2 = 36\).
- Test each given equation against the requirements; only \((x - 6)^2 + y^2 = 36\) and \((x + 6)^2 + y^2 = 36\) fulfill both conditions.
- Final answer: \(\boxed{(x-6)^2 + y^2 = 36, (x+6)^2 + y^2 = 36}\).
Explanation
- Determine the general equation of the circle
A circle is defined by its center \((h, k)\) and its radius \(r\). The general standard form of the equation of a circle is \((x - h)^2 + (y - k)^2 = r^2\). Given that the circle has a diameter of \(12\) units, its radius \(r\) is half the diameter, which is \(r = \frac{12}{2} = 6\). Consequently, the square of the radius is \(r^2 = 6^2 = 36\). The problem states that the center of the circle lies on the \(x\)-axis, meaning its \(y\)-coordinate \(k\) is \(0\). Thus, any circle meeting these conditions must satisfy the equation \((x - h)^2 + y^2 = 36\) for some real number \(h\).
- Evaluate the given options
We examine the given options to see which fit the derived form \((x - h)^2 + y^2 = 36\):
- \((x-12)^2 + y^2 = 12\): Here, \(r^2 = 12
eq 36\). This does not represent a circle with radius 6.
- \((x-6)^2 + y^2 = 36\): This fits the form with \(h = 6\). This is a valid equation.
- \(x^2 + y^2 = 12\): Here, \(r^2 = 12
eq 36\). This does not represent a circle with radius 6.
- \(x^2 + y^2 = 144\): Here, \(r^2 = 144
eq 36\). This does not represent a circle with radius 6.
- \((x+6)^2 + y^2 = 36\): This fits the form with \(h = -6\). This is a valid equation.
- \((x+12)^2 + y^2 = 144\): Here, \(r^2 = 144
eq 36\). This does not represent a circle with radius 6.
- Final conclusion
Based on the evaluation, the equations that represent a circle with a diameter of 12 and a center on the \(x\)-axis are \((x-6)^2 + y^2 = 36\) and \((x+6)^2 + y^2 = 36\). The final answer is \(\boxed{(x-6)^2 + y^2 = 36, (x+6)^2 + y^2 = 36}\).
Examples
In computer-aided design (CAD) software, understanding how to express shapes mathematically is essential for precision engineering. If a circular gear component with a diameter of 12 cm needs to be positioned along a central track (the x-axis), an engineer uses the circle equation \((x-h)^2 + y^2 = 36\) to define its boundaries. By varying \(h\), the engineer can move the gear's center along the track while maintaining its size, demonstrating the practical utility of coordinate geometry in manufacturing and design.