Answer :
To solve the problem of identifying the correct equation for a circle with a given radius [tex]\( r \)[/tex] and center at [tex]\( (h, v) \)[/tex], let's recall the standard form of a circle's equation. The standard equation for a circle with a radius [tex]\( r \)[/tex] and a center at [tex]\( (h, v) \)[/tex] is derived as follows:
1. Start with the definition of a circle:
A circle is the set of all points [tex]\((x, y)\)[/tex] in a plane that are a fixed distance [tex]\( r \)[/tex] (the radius) from a fixed point [tex]\((h, v)\)[/tex] (the center).
2. Use the distance formula:
The distance between any point [tex]\((x, y)\)[/tex] on the circle and the center [tex]\((h, v)\)[/tex] is [tex]\( r \)[/tex]. The distance formula is:
[tex]\[ \sqrt{(x - h)^2 + (y - v)^2} = r \][/tex]
3. Square both sides to eliminate the square root:
[tex]\[ (x - h)^2 + (y - v)^2 = r^2 \][/tex]
This is the equation of a circle centered at [tex]\((h, v)\)[/tex] with radius [tex]\( r \)[/tex].
Now, let's compare this to the options given in the question:
- Option A: [tex]\((x-h)^2+(y-v)^2=r^2\)[/tex]
- Option B: [tex]\((x+h)^2+(y+v)^2=r^2\)[/tex]
- Option C: [tex]\((x-v)^2+(y-h)^2=r^2\)[/tex]
- Option D: [tex]\(h^2+v^2=r^2\)[/tex]
From our derivation, we see that option A [tex]\((x-h)^2+(y-v)^2=r^2\)[/tex] is the standard form of the circle's equation, matching perfectly with the one we derived.
Hence, the correct answer is:
A. [tex]\((x-h)^2+(y-v)^2=r^2\)[/tex]
1. Start with the definition of a circle:
A circle is the set of all points [tex]\((x, y)\)[/tex] in a plane that are a fixed distance [tex]\( r \)[/tex] (the radius) from a fixed point [tex]\((h, v)\)[/tex] (the center).
2. Use the distance formula:
The distance between any point [tex]\((x, y)\)[/tex] on the circle and the center [tex]\((h, v)\)[/tex] is [tex]\( r \)[/tex]. The distance formula is:
[tex]\[ \sqrt{(x - h)^2 + (y - v)^2} = r \][/tex]
3. Square both sides to eliminate the square root:
[tex]\[ (x - h)^2 + (y - v)^2 = r^2 \][/tex]
This is the equation of a circle centered at [tex]\((h, v)\)[/tex] with radius [tex]\( r \)[/tex].
Now, let's compare this to the options given in the question:
- Option A: [tex]\((x-h)^2+(y-v)^2=r^2\)[/tex]
- Option B: [tex]\((x+h)^2+(y+v)^2=r^2\)[/tex]
- Option C: [tex]\((x-v)^2+(y-h)^2=r^2\)[/tex]
- Option D: [tex]\(h^2+v^2=r^2\)[/tex]
From our derivation, we see that option A [tex]\((x-h)^2+(y-v)^2=r^2\)[/tex] is the standard form of the circle's equation, matching perfectly with the one we derived.
Hence, the correct answer is:
A. [tex]\((x-h)^2+(y-v)^2=r^2\)[/tex]
