To determine which equation represents a circle that contains the point [tex]$(-5, -3)$[/tex] and has a center at [tex]$(-2, 1)$[/tex], we need to follow these steps closely:
1. Identify the coordinates of the center and the point on the circle:
- Center: [tex]$(-2, 1)$[/tex]
- Point on the circle: [tex]$(-5, -3)$[/tex]
2. Calculate the radius of the circle using the distance formula:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Substituting in the given points:
[tex]\[
\text{Distance} = \sqrt{((-5) - (-2))^2 + ((-3) - 1)^2}
\][/tex]
Simplify the expressions inside the square root:
[tex]\[
= \sqrt{(-5 + 2)^2 + (-3 - 1)^2}
\][/tex]
[tex]\[
= \sqrt{(-3)^2 + (-4)^2}
\][/tex]
[tex]\[
= \sqrt{9 + 16}
\][/tex]
[tex]\[
= \sqrt{25}
\][/tex]
[tex]\[
= 5
\][/tex]
Thus, the radius [tex]\( r \)[/tex] is 5.
3. Determine the radius squared for the equation of the circle:
[tex]\[
r^2 = 5^2 = 25
\][/tex]
4. Match the center and radius squared to the given equations of circles:
- [tex]$(x-1)^2+(y+2)^2=25$[/tex] has center [tex]$(1, -2)$[/tex] and radius squared [tex]$25$[/tex].
- [tex]$(x+2)^2+(y-1)^2=5$[/tex] has center [tex]$(-2, 1)$[/tex] but radius squared [tex]$5$[/tex].
- [tex]$(x+2)^2+(y-1)^2=25$[/tex] has center [tex]$(-2, 1)$[/tex] and radius squared [tex]$25$[/tex].
- [tex]$(x-1)^2+(y+2)^2=5$[/tex] has center [tex]$(1, -2)$[/tex] and radius squared [tex]$5$[/tex].
5. Select the correct equation:
The circle with center [tex]$(-2, 1)$[/tex] and radius squared [tex]$25$[/tex] is represented by the equation:
[tex]\[
(x+2)^2+(y-1)^2=25
\][/tex]
Therefore, the equation which represents a circle that contains the point [tex]$(-5, -3)$[/tex] and has a center at [tex]$(-2, 1)$[/tex] is:
[tex]\[
(x+2)^2+(y-1)^2=25
\][/tex]
