Question

write the standard form of the equation of the circle with the given characteristics. center: (-2, -9); solution point: (1, -13)

Explanation:

Step1: Recall the standard form of a circle's equation

The standard form of the equation of a circle with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\). We know the center \((h, k)=(-2, -9)\), so we can substitute these values into the equation: \((x - (-2))^2 + (y - (-9))^2 = r^2\), which simplifies to \((x + 2)^2 + (y + 9)^2 = r^2\).

Step2: Find the radius \(r\)

We know that the point \((1, -13)\) lies on the circle, so we can substitute \(x = 1\) and \(y = -13\) into the equation from Step 1 to find \(r^2\).
Substituting, we get \((1 + 2)^2 + (-13 + 9)^2 = r^2\).
First, calculate the values inside the parentheses: \(1 + 2 = 3\) and \(-13 + 9 = -4\).
Then, square these values: \(3^2 = 9\) and \((-4)^2 = 16\).
Add these results: \(9 + 16 = r^2\), so \(r^2 = 25\).

Step3: Write the final equation

Now that we know \(h = -2\), \(k = -9\), and \(r^2 = 25\), we substitute these into the standard form equation: \((x + 2)^2 + (y + 9)^2 = 25\).

Answer:

\((x + 2)^2 + (y + 9)^2 = 25\)