Answer :
Let's examine the given equation of the circle and interpret its components to determine the accurate characteristics of the circle.
The equation of the circle provided is:
[tex]$(x - 5)^2 + (y + 7)^2 = 49$[/tex]
This equation is in the standard form of a circle's equation:
[tex]$(x - h)^2 + (y - k)^2 = r^2,$[/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
### Step-by-Step Solution:
1. Identify the center coordinates (h, k):
- By comparing the given equation [tex]\((x - 5)^2 + (y + 7)^2 = 49\)[/tex] with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can see that:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = -7\)[/tex]
Therefore, the center of the circle is [tex]\((5, -7)\)[/tex].
2. Identify the radius:
- The right-hand side of the equation is [tex]\(49\)[/tex]. According to the standard form, this represents [tex]\(r^2\)[/tex].
- So, the radius [tex]\(r\)[/tex] is the square root of [tex]\(49\)[/tex]:
[tex]$r = \sqrt{49} = 7.$[/tex]
Therefore, the true statement about the circle is:
- The circle is centered at [tex]\((5, -7)\)[/tex] and has a radius of [tex]\(7\)[/tex].
Given the choices:
- The circle is centered at [tex]\((-5, 7)\)[/tex] and has a radius of 7.
- The circle is centered at [tex]\((5, -7)\)[/tex] and has a diameter of 7.
- The circle is centered at [tex]\((5, -7)\)[/tex] and has a radius of 7.
- The circle is centered at [tex]\((-5, 7)\)[/tex] and has a diameter of 7.
The correct choice is:
- The circle is centered at [tex]\((5, -7)\)[/tex] and has a radius of 7.
The equation of the circle provided is:
[tex]$(x - 5)^2 + (y + 7)^2 = 49$[/tex]
This equation is in the standard form of a circle's equation:
[tex]$(x - h)^2 + (y - k)^2 = r^2,$[/tex]
where [tex]\((h, k)\)[/tex] is the center of the circle and [tex]\(r\)[/tex] is the radius.
### Step-by-Step Solution:
1. Identify the center coordinates (h, k):
- By comparing the given equation [tex]\((x - 5)^2 + (y + 7)^2 = 49\)[/tex] with the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can see that:
- [tex]\(h = 5\)[/tex]
- [tex]\(k = -7\)[/tex]
Therefore, the center of the circle is [tex]\((5, -7)\)[/tex].
2. Identify the radius:
- The right-hand side of the equation is [tex]\(49\)[/tex]. According to the standard form, this represents [tex]\(r^2\)[/tex].
- So, the radius [tex]\(r\)[/tex] is the square root of [tex]\(49\)[/tex]:
[tex]$r = \sqrt{49} = 7.$[/tex]
Therefore, the true statement about the circle is:
- The circle is centered at [tex]\((5, -7)\)[/tex] and has a radius of [tex]\(7\)[/tex].
Given the choices:
- The circle is centered at [tex]\((-5, 7)\)[/tex] and has a radius of 7.
- The circle is centered at [tex]\((5, -7)\)[/tex] and has a diameter of 7.
- The circle is centered at [tex]\((5, -7)\)[/tex] and has a radius of 7.
- The circle is centered at [tex]\((-5, 7)\)[/tex] and has a diameter of 7.
The correct choice is:
- The circle is centered at [tex]\((5, -7)\)[/tex] and has a radius of 7.