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Answer :
To determine which equation describes a concentric circle to the given circle, let's analyze the equations.
1. Understand the given circle:
The equation of the circle given is [tex]\(x^2 + y^2 = 169\)[/tex]. This is the standard form of a circle equation centered at the origin [tex]\((0, 0)\)[/tex] with a radius [tex]\(r\)[/tex] where [tex]\(r^2 = 169\)[/tex]. Thus, the radius [tex]\(r\)[/tex] is [tex]\(13\)[/tex].
2. Definition of concentric circles:
Concentric circles share the same center. This means that if the original circle's center is at the origin [tex]\((0, 0)\)[/tex], any concentric circle must also be centered at [tex]\((0, 0)\)[/tex], but it can have a different radius.
3. Evaluate the options:
- Option A: [tex]\((x-2)^2 + (y-3)^2 = 169\)[/tex]
This circle is centered at [tex]\((2, 3)\)[/tex], so it's not concentric with the given circle.
- Option B: [tex]\((x+3)^2 + (y+5)^2 = 169\)[/tex]
This circle is centered at [tex]\((-3, -5)\)[/tex], so it's also not concentric with the given circle.
- Option C: [tex]\(x^2 + y^2 = 64\)[/tex]
This circle is centered at the origin [tex]\((0, 0)\)[/tex]. It maintains the same center as the given circle but has a different radius. Therefore, this is a concentric circle.
- Option D: [tex]\((x-2)^2 + (y-0)^2 = 169\)[/tex]
This circle is centered at [tex]\((2, 0)\)[/tex], which means it is not concentric with the given circle.
Based on this analysis, Option C: [tex]\(x^2 + y^2 = 64\)[/tex] is the equation that describes a concentric circle to the given circle, as it maintains the same center at the origin.
1. Understand the given circle:
The equation of the circle given is [tex]\(x^2 + y^2 = 169\)[/tex]. This is the standard form of a circle equation centered at the origin [tex]\((0, 0)\)[/tex] with a radius [tex]\(r\)[/tex] where [tex]\(r^2 = 169\)[/tex]. Thus, the radius [tex]\(r\)[/tex] is [tex]\(13\)[/tex].
2. Definition of concentric circles:
Concentric circles share the same center. This means that if the original circle's center is at the origin [tex]\((0, 0)\)[/tex], any concentric circle must also be centered at [tex]\((0, 0)\)[/tex], but it can have a different radius.
3. Evaluate the options:
- Option A: [tex]\((x-2)^2 + (y-3)^2 = 169\)[/tex]
This circle is centered at [tex]\((2, 3)\)[/tex], so it's not concentric with the given circle.
- Option B: [tex]\((x+3)^2 + (y+5)^2 = 169\)[/tex]
This circle is centered at [tex]\((-3, -5)\)[/tex], so it's also not concentric with the given circle.
- Option C: [tex]\(x^2 + y^2 = 64\)[/tex]
This circle is centered at the origin [tex]\((0, 0)\)[/tex]. It maintains the same center as the given circle but has a different radius. Therefore, this is a concentric circle.
- Option D: [tex]\((x-2)^2 + (y-0)^2 = 169\)[/tex]
This circle is centered at [tex]\((2, 0)\)[/tex], which means it is not concentric with the given circle.
Based on this analysis, Option C: [tex]\(x^2 + y^2 = 64\)[/tex] is the equation that describes a concentric circle to the given circle, as it maintains the same center at the origin.
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