Sure! Let's solve this step-by-step.
1. Identify the Given Data:
- Center of the circle: [tex]\((h, k) = (2, 4)\)[/tex]
- Point on the circle: [tex]\((x_1, y_1) = (8, 6)\)[/tex]
2. Calculate the Radius:
- The radius of the circle is the distance between the center [tex]\((2, 4)\)[/tex] and the point [tex]\((8, 6)\)[/tex].
- The distance formula between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
- Plugging in the points [tex]\((2, 4)\)[/tex] and [tex]\((8, 6)\)[/tex]:
[tex]\[
\text{Radius} = \sqrt{(8 - 2)^2 + (6 - 4)^2}
\][/tex]
[tex]\[
\text{Radius} = \sqrt{6^2 + 2^2}
\][/tex]
[tex]\[
\text{Radius} = \sqrt{36 + 4} = \sqrt{40} \approx 6.324555320336759
\][/tex]
3. Square the Radius:
- The standard equation of a circle involves the radius squared.
- Therefore, we need to square the radius value we just calculated:
[tex]\[
\text{Radius}^2 = (\sqrt{40})^2
\][/tex]
[tex]\[
\text{Radius}^2 = 40.00000000000001
\][/tex]
4. Write the Standard Equation of the Circle:
- The standard form for the 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]
- With [tex]\(h = 2\)[/tex], [tex]\(k = 4\)[/tex], and [tex]\(r^2 = 40.00000000000001\)[/tex]:
[tex]\[
(x - 2)^2 + (y - 4)^2 = 40.00000000000001
\][/tex]
5. Write the Final Answer:
- The standard equation of the circle with center [tex]\((2, 4)\)[/tex] passing through the point [tex]\((8, 6)\)[/tex] is:
[tex]\[
(x - 2)^2 + (y - 4)^2 = 40.00000000000001
\][/tex]
This is the required equation of the circle.
