Answer :
To determine the equation of a circle centered at the origin with a given radius, we can use the general formula for the equation of a circle.
The general equation of a circle centered at [tex]\((h, k)\)[/tex] with radius [tex]\(r\)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Since the circle is centered at the origin, [tex]\((h, k) = (0, 0)\)[/tex]. Given the radius [tex]\(r = 3\)[/tex], we can substitute [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r\)[/tex] into the formula.
1. Substitute [tex]\(h = 0\)[/tex] and [tex]\(k = 0\)[/tex] into the equation:
[tex]\[ (x - 0)^2 + (y - 0)^2 = r^2 \][/tex]
2. Given that the radius [tex]\(r = 3\)[/tex], substitute [tex]\(r = 3\)[/tex] into the equation:
[tex]\[ (x - 0)^2 + (y - 0)^2 = 3^2 \][/tex]
3. Simplify the terms:
[tex]\[ (x - 0)^2 = x^2 \][/tex]
[tex]\[ (y - 0)^2 = y^2 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
4. Putting it all together, we get:
[tex]\[ x^2 + y^2 = 9 \][/tex]
Therefore, the equation of the circle centered at the origin with a radius of 3 is:
[tex]\[ x^2 + y^2 = 9 \][/tex]
The general equation of a circle centered at [tex]\((h, k)\)[/tex] with radius [tex]\(r\)[/tex] is:
[tex]\[ (x - h)^2 + (y - k)^2 = r^2 \][/tex]
Since the circle is centered at the origin, [tex]\((h, k) = (0, 0)\)[/tex]. Given the radius [tex]\(r = 3\)[/tex], we can substitute [tex]\(h\)[/tex], [tex]\(k\)[/tex], and [tex]\(r\)[/tex] into the formula.
1. Substitute [tex]\(h = 0\)[/tex] and [tex]\(k = 0\)[/tex] into the equation:
[tex]\[ (x - 0)^2 + (y - 0)^2 = r^2 \][/tex]
2. Given that the radius [tex]\(r = 3\)[/tex], substitute [tex]\(r = 3\)[/tex] into the equation:
[tex]\[ (x - 0)^2 + (y - 0)^2 = 3^2 \][/tex]
3. Simplify the terms:
[tex]\[ (x - 0)^2 = x^2 \][/tex]
[tex]\[ (y - 0)^2 = y^2 \][/tex]
[tex]\[ 3^2 = 9 \][/tex]
4. Putting it all together, we get:
[tex]\[ x^2 + y^2 = 9 \][/tex]
Therefore, the equation of the circle centered at the origin with a radius of 3 is:
[tex]\[ x^2 + y^2 = 9 \][/tex]
