- Determine the radius \(r\) from the diameter \(d=12\), finding \(r=6\), so \(r^2=36\).
- Identify that the center on the \(y\)-axis implies the equation form \(x^2 + (y-k)^2 = 36\).
- Match this form against the given choices, identifying:
- \(x^2 + (y - 3)^2 = 36\)
- \(x^2 + (y + 8)^2 = 36\)
- The final answer is \(\boxed{x^2 + (y - 3)^2 = 36, x^2 + (y + 8)^2 = 36}\).
Explanation
- Analyze the requirements for the circle equation.
To identify the correct equations for a circle, we first use the standard form of a circle's equation: \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center of the circle and \(r\) is its radius.
Given that the diameter of the circle is \(12\) units, we calculate the radius as:
\(r = \frac{d}{2} = \frac{12}{2} = 6\).
The square of the radius, which appears in the standard equation as \(r^2\), is therefore:
\(r^2 = 6^2 = 36\).
The problem also states that the center of the circle lies on the \(y\)-axis. Any point on the \(y\)-axis has an \(x\)-coordinate of \(0\). Thus, the center \((h, k)\) must have \(h = 0\). Substituting \(h = 0\) into the standard equation, we get:
\((x - 0)^2 + (y - k)^2 = 36\)
\(x^2 + (y - k)^2 = 36\)
This confirms that any equation representing such a circle must have \(x^2\) as the first term and \(36\) as the constant on the right side.
- Evaluate the given options.
We evaluate the provided options against our derived criteria: \(x^2 + (y - k)^2 = 36\).
- \(x^2 + (y - 3)^2 = 36\): This matches the form \(x^2 + (y - k)^2 = 36\) with \(k=3\). This is a valid equation.
- \(x^2 + (y - 5)^2 = 6\): This does not match the required constant \(r^2 = 36\).
- \((x - 4)^2 + y^2 = 36\): This does not match the requirement \(h=0\), as here \(h=4\).
- \((x + 6)^2 + y^2 = 144\): This does not match either the required \(h=0\) nor the required \(r^2=36\).
- \(x^2 + (y + 8)^2 = 36\): This matches the form \(x^2 + (y - k)^2 = 36\) with \(k=-8\). This is a valid equation.
Comparing the options, we identify the two correct equations.
- State the final answer.
Based on our analysis, the two equations that represent circles with a diameter of \(12\) and a center on the \(y\)-axis are:
\(x^2 + (y - 3)^2 = 36\) and \(x^2 + (y + 8)^2 = 36\).
The final answer is:
\(\boxed{x^2 + (y - 3)^2 = 36, x^2 + (y + 8)^2 = 36}\)
Examples
Understanding the equation of a circle is essential for fields like computer graphics and urban planning. For example, if a city planner needs to design a circular park with a diameter of 12 units centered at a specific location on a major avenue (represented by the y-axis), they can use these equations to map the boundaries of the park precisely. Setting the equation as \(x^2 + (y - k)^2 = 36\) allows them to shift the center of the park along the avenue by simply changing the value of \(k\), ensuring the park's size remains consistent while its location is flexible.