Question

1. - / 1 points write the standard form of the equation of the circle with the given characteristics. center: (-2, -6); solution point: (2, -9)

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 \((h, k)=(-2, -6)\), so we need to find the radius \(r\). The radius is the distance between the center \((-2, -6)\) and the solution point \((2, -9)\).

Step2: Calculate the radius using the distance formula

The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). Here, \(x_1=-2\), \(y_1 = - 6\), \(x_2=2\), \(y_2=-9\).
\[

$$\begin{align*} r&=\sqrt{(2 - (-2))^2+(-9 - (-6))^2}\\ &=\sqrt{(2 + 2)^2+(-9 + 6)^2}\\ &=\sqrt{4^2+(-3)^2}\\ &=\sqrt{16 + 9}\\ &=\sqrt{25}\\ & = 5 \end{align*}$$

\]

Step3: Substitute \(h\), \(k\) and \(r\) into the circle's equation

We have \(h=-2\), \(k = - 6\) and \(r = 5\). Substituting into \((x - h)^2+(y - k)^2=r^2\):
\[
(x - (-2))^2+(y - (-6))^2=5^2
\]
Simplify the equation:
\[
(x + 2)^2+(y + 6)^2=25
\]

Answer:

\((x + 2)^2+(y + 6)^2 = 25\)