- The standard form for the equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).
- Substituting the given center \((2, -8)\) and radius \(11\) into the formula gives \((x - 2)^2 + (y - (-8))^2 = 11^2\).
- Simplifying this expression results in the final equation \((x - 2)^2 + (y + 8)^2 = 121\).
- The final answer is \(\boxed{(x - 2)^2 + (y + 8)^2 = 121}\).
Explanation
- Understand the Standard Form of a Circle's Equation.
The equation of a circle with center \((h, k)\) and radius \(r\) is given by the standard form equation:
\[(x - h)^2 + (y - k)^2 = r^2\]
From the problem, we are given:
- Center \((h, k) = (2, -8)\)
- Radius \(r = 11\)
We need to substitute these values into the standard form.
- Substitute and Simplify the Equation.
Substituting the values \(h = 2\), \(k = -8\), and \(r = 11\) into the standard form:
\[(x - 2)^2 + (y - (-8))^2 = 11^2\]
Next, simplify the terms inside the parentheses and the value on the right side:
- For the \(y\)-term: \(y - (-8) = y + 8\)
- For the radius squared: \(11^2 = 121\)
Thus, the equation becomes:
\[(x - 2)^2 + (y + 8)^2 = 121\]
- Select the Correct Option.
Now we compare our derived equation, \((x - 2)^2 + (y + 8)^2 = 121\), with the provided options:
- Option 1: \((x - 8)^2 + (y + 2)^2 = 11\)
- Option 2: \((x - 2)^2 + (y + 8)^2 = 121\)
- Option 3: \((x + 2)^2 + (y - 8)^2 = 11\)
- Option 4: \((x + 8)^2 + (y - 2)^2 = 121\)
Our derived equation matches the second option exactly. Therefore, the correct equation for the circle is \((x - 2)^2 + (y + 8)^2 = 121\).
Examples
Understanding the equation of a circle is fundamental in fields like civil engineering and architecture. For example, if a civil engineer is designing a circular roundabout with its center located at coordinate \((2, -8)\) on a grid and requires a radius of \(11\) meters, they can use the circle equation \((x - 2)^2 + (y + 8)^2 = 121\) to model the perimeter of the road, ensuring that the layout is perfectly circular and fits precisely into the planned location.