Answer:
[tex](x - 3)^2 + (y - 6)^2 = 10[/tex]
Step-by-step explanation:
To determine the equation of a circle, we need the coordinates of its center and one point on the circumference. The general form of the equation of a circle with center [tex](h, k)[/tex] and radius [tex]r[/tex] is:
[tex] \Large\boxed{\boxed{(x - h)^2 + (y - k)^2 = r^2}}[/tex]
Given the center of the circle is [tex](3, 6)[/tex] and a point on the circumference is [tex](4, 9)[/tex], we can plug these values into the equation to solve for the radius ([tex]r[/tex]).
Let's use the coordinates of the center and the point on the circumference:
- Center: [tex](h, k) = (3, 6)[/tex]
- Circumference point: [tex](x, y) = (4, 9)[/tex]
We substitute these values into the equation:
[tex](4 - 3)^2 + (9 - 6)^2 = r^2[/tex]
[tex]1^2 + 3^2 = r^2[/tex]
[tex]1 + 9 = r^2[/tex]
[tex]10 = r^2[/tex]
So, the radius ([tex]r[/tex]) of the circle is [tex]r = \sqrt{10}[/tex].
Now, we can write the equation of the circle using the center and the radius:
[tex](x - 3)^2 + (y - 6)^2 = (\sqrt{10})^2[/tex]
[tex](x - 3)^2 + (y - 6)^2 = 10[/tex]
Thus, the equation of the circle with a center at [tex](3, 6)[/tex] and passing through the point [tex](4, 9)[/tex] on the circumference is:
[tex] \Large\boxed{\boxed{(x - 3)^2 + (y - 6)^2 = 10}}[/tex]
