To determine which equation represents a circle centered at [tex]\((3,5)\)[/tex] and passing through the point [tex]\((-2,9)\)[/tex], let's follow the steps for finding the equation of a circle in standard form and verifying that point:
1. Identify the center [tex]\((h, k)\)[/tex]:
The center of the circle is given as [tex]\((3, 5)\)[/tex].
2. Point on the circle [tex]\((x_1, y_1)\)[/tex]:
The point on the circle is [tex]\((-2, 9)\)[/tex].
3. Calculate the radius squared:
Use the distance formula to find the radius squared, which in turn will help us derive the equation of the circle.
[tex]\[
\begin{aligned}
r^2 &= (x_1 - h)^2 + (y_1 - k)^2 \\
& = (-2 - 3)^2 + (9 - 5)^2 \\
& = (-5)^2 + (4)^2 \\
& = 25 + 16 \\
& = 41.
\end{aligned}
\][/tex]
4. Formulate the equation of the circle:
The standard form of the equation of a circle with center [tex]\((h, k)\)[/tex] and radius squared [tex]\(r^2\)[/tex] is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
Plugging in the values [tex]\(h = 3\)[/tex], [tex]\(k = 5\)[/tex], and [tex]\(r^2 = 41\)[/tex]:
[tex]\[
(x - 3)^2 + (y - 5)^2 = 41.
\][/tex]
Now, let's match this with the options provided:
- Option A: [tex]\((x - 3)^2 + (y - 5)^2 = 41\)[/tex] – This matches our derived equation.
- Option B: [tex]\((x + 3)^2 + (y + 5)^2 = 17\)[/tex] – Clearly, this doesn't match the center or radius squared.
- Option C: [tex]\((x + 3)^2 + (y + 5)^2 = 41\)[/tex] – This also doesn't match the center.
- Option D: [tex]\((x - 3)^2 + (y - 5)^2 = 17\)[/tex] – Only the center matches, but the radius squared is incorrect.
Thus, the correct equation that represents the circle centered at [tex]\((3, 5)\)[/tex] and passing through the point [tex]\((-2, 9)\)[/tex] is:
[tex]\[
\boxed{(x - 3)^2 + (y - 5)^2 = 41}
\][/tex] and the correct choice is Option A.
