Sure, let's solve this step-by-step:
### Part A: Center at (0,0) and a point on the circle at (0,6)
1. Identify the center and a point on the circle:
- Center: [tex]\((0, 0)\)[/tex]
- Point on the circle: [tex]\((0, 6)\)[/tex]
2. Calculate the radius of the circle:
- The radius is the distance from the center to the point on the circle.
- Using the distance formula: [tex]\( \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]
- Substituting the values: [tex]\(\text{distance} = \sqrt{(0 - 0)^2 + (6 - 0)^2} = \sqrt{0 + 36} = \sqrt{36} = 6\)[/tex]
3. Write the equation of the circle:
- The general equation of a circle is: [tex]\( (x - h)^2 + (y - k)^2 = r^2 \)[/tex]
- Here, [tex]\(h = 0\)[/tex], [tex]\(k = 0\)[/tex], and [tex]\(r = 6\)[/tex]
- Substituting the values, we get: [tex]\( (x - 0)^2 + (y - 0)^2 = 6^2 \)[/tex]
- Simplifying: [tex]\( (x - 0)^2 + (y - 0)^2 = 36 \)[/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 0)^2 + (y - 0)^2 = 36
\][/tex]
### Part B: Center at (1,2) and a point on the circle at (4,2)
1. Identify the center and a point on the circle:
- Center: [tex]\((1, 2)\)[/tex]
- Point on the circle: [tex]\((4, 2)\)[/tex]
2. Calculate the radius of the circle:
- The radius is the distance from the center to the point on the circle.
- Using the distance formula: [tex]\( \text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \)[/tex]
- Substituting the values: [tex]\(\text{distance} = \sqrt{(4 - 1)^2 + (2 - 2)^2} = \sqrt{3^2 + 0} = \sqrt{9} = 3\)[/tex]
3. Write the equation of the circle:
- The general equation of a circle is: [tex]\( (x - h)^2 + (y - k)^2 = r^2 \)[/tex]
- Here, [tex]\(h = 1\)[/tex], [tex]\(k = 2\)[/tex], and [tex]\(r = 3\)[/tex]
- Substituting the values, we get: [tex]\( (x - 1)^2 + (y - 2)^2 = 3^2 \)[/tex]
- Simplifying: [tex]\( (x - 1)^2 + (y - 2)^2 = 9 \)[/tex]
Therefore, the equation of the circle is:
[tex]\[
(x - 1)^2 + (y - 2)^2 = 9
\][/tex]
To summarize:
- For part (a), the equation of the circle is [tex]\( (x - 0)^2 + (y - 0)^2 = 36 \)[/tex].
- For part (b), the equation of the circle is [tex]\( (x - 1)^2 + (y - 2)^2 = 9 \)[/tex].
