https://file.aiquickdraw.com/mathbot/file/172705637010542h2koqd.jpg

The problem asks to find the equation of a circle centered at point \( A \) and passing through point \( K \). 

### Step-by-step Solution:

1. **Identify the given points:**
   - Point \( A \) (center of the circle) is \( (2, -1) \).
   - Point \( K \), which lies on the circle, is \( (8, 5) \).

2. **Determine the radius:**
   The radius \( r \) of the circle can be found by calculating the distance between points \( A \) and \( K \). Using the distance formula:
   
   \[
   r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
   \]

   Substituting \( (x_1, y_1) = (2, -1) \) and \( (x_2, y_2) = (8, 5) \):

   \[
   r = \sqrt{(8 - 2)^2 + (5 - (-1))^2} = \sqrt{6^2 + 6^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
   \]

3. **Equation of the circle:**
   The standard equation of a circle with center \( (h, k) \) and radius \( r \) is:

   \[
   (x - h)^2 + (y - k)^2 = r^2
   \]

   Substituting the center \( (2, -1) \) and radius \( 6\sqrt{2} \):

   \[
   (x - 2)^2 + (y + 1)^2 = (6\sqrt{2})^2
   \]

   Simplifying:

   \[
   (x - 2)^2 + (y + 1)^2 = 72
   \]

### Final Answer:
The equation of the circle is:

\[
(x - 2)^2 + (y + 1)^2 = 72
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you derive the formula for the distance between two points?
2. How can you find the center of a circle given its equation?
3. What are the steps to transform a general circle equation into its standard form?
4. How do you determine if a point lies inside, on, or outside a circle given its equation?
5. Can you find the equation of a circle if two points on the circle are given and the circle passes through the origin?

### Tip:
Always double-check the coordinates and calculations when dealing with geometric problems involving distance and equations to avoid simple errors.

Dari ilustrasi di atas, tentukan persamaan lingkaran yang berpusat di titik A dan melalui titik K.

Pelajari cara menentukan persamaan lingkaran yang berpusat di titik A (2,-1) dan melalui titik K (8,5) dengan solusi langkah demi langkah. Termasuk konsep geometri lingkaran dan penggunaan rumus jarak.

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