- Use the center \((-2, 1)\) to set up the equation \((x+2)^2 + (y-1)^2 = r^2\).
- Calculate the radius squared by substituting the point \((-5, -3)\) into the equation to find \(r^2 = 25\).
- State the final circle equation as \((x + 2)^2 + (y - 1)^2 = 25\).
- The correct option is \((x + 2)^2 + (y - 1)^2 = 25\).
Explanation
- Identify the standard equation of the circle.
The standard equation of a circle with center \((h, k)\) and radius \(r\) is given by \((x - h)^2 + (y - k)^2 = r^2\). Given the center of the circle is \((-2, 1)\), we substitute \(h = -2\) and \(k = 1\) into the standard equation:
\((x - (-2))^2 + (y - 1)^2 = r^2\)
This simplifies to:
\((x + 2)^2 + (y - 1)^2 = r^2\)
- Calculate the radius squared of the circle.
The circle passes through the point \((-5, -3)\). To find the radius squared (\(r^2\)), we substitute \(x = -5\) and \(y = -3\) into the equation derived in the previous step:
\((-5 + 2)^2 + (-3 - 1)^2 = r^2\)
\((-3)^2 + (-4)^2 = r^2\)
Evaluating the squares:
\(9 + 16 = r^2\)
\(r^2 = 25\)
- Formulate the final equation of the circle.
Now that we have the center \((-2, 1)\) and the radius squared \(r^2 = 25\), we can write the final equation of the circle by substituting these values into the standard form:
\((x + 2)^2 + (y - 1)^2 = 25\)
Comparing this with the given options, we can see it matches one of them.
Examples
Circles are used in many real-world applications, such as in radio wave coverage, where a transmitter at a certain location (the center) covers a certain area (the radius). If a radio tower is located at \((-2, 1)\) and it can reach a point at \((-5, -3)\), the equation of the boundary of the coverage area is \((x + 2)^2 + (y - 1)^2 = 25\), where 25 is the square of the transmission radius. This helps in understanding the range of signal transmission in urban planning.