Answer :
Answer:
[tex](x + 1)^{2} + (y + 2)^{2} = 25[/tex].
Step-by-step explanation:
If the center of a circle is [tex](x_{0},\, y_{0})[/tex], and the radius of the circle is [tex]r[/tex], the equation of the circle would be:
[tex](x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}[/tex].
In this question:
- The center of the circle is point [tex]{\sf A}[/tex]. The coordinates of this point are given: [tex]x_{0} = (-1)[/tex], [tex]y_{0} = (-2)[/tex],
- It is given that the radius of this circle is equal to the length of the segment [tex]{\sf AC}[/tex]. While the exact length of this segment is not given, the coordinates of [tex]{\sf A}[/tex] and [tex]{\sf C}[/tex] are given.
To find the length of this segment, apply the distance formula: the distance between two points in a cartesian plane, [tex](x_{0},\, y_{0})[/tex] and [tex](x_{1},\, y_{1})[/tex], is:
[tex]\displaystyle \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2}}[/tex].
Using this formula, the length of segment [tex]{\sf AC}[/tex] would be:
[tex]\begin{aligned} & \sqrt{(x_{1} - x_{0})^{2} + (y_{1} - y_{0})^{2}} \\ =\; & \sqrt{((-5) - (-1))^{2} + ((1 - (-2))^{2}} \\ =\; & \sqrt{25}\\ =\; & 5\end{aligned}[/tex],
The radius of the circle is equal to the length of this segment.
Hence, the equation for this circle would be:
[tex](x - x_{0})^{2} + (y - y_{0})^{2} = r^{2}[/tex].
[tex](x - (-1))^{2} + (y - (-2))^{2} = 5^{2}[/tex].
[tex](x + 1)^{2} + (y + 2)^{2} = 25[/tex].
