Find the equation for the circle with center 

​(5,3​) 

and passing through 

​(-1,−4​).

To find the equation of a circle with a given center and a point on the circle, we can use the standard form of the equation of a circle:

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

where \((h, k)\) is the center of the circle and \(r\) is the radius.

Given:
- Center: \((h, k) = (5, 3)\)
- Point on the circle: \((-1, -4)\)

### Step 1: Find the radius \(r\)
The radius \(r\) is the distance between the center of the circle and the point on the circle. We can use the distance formula to calculate this:

\[
r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

Substituting the given points:

\[
r = \sqrt{(-1 - 5)^2 + (-4 - 3)^2} = \sqrt{(-6)^2 + (-7)^2} = \sqrt{36 + 49} = \sqrt{85}
\]

### Step 2: Write the equation of the circle
Now that we know the radius \(r = \sqrt{85}\), we can substitute the center \((5, 3)\) and the radius into the equation of the circle:

\[
(x - 5)^2 + (y - 3)^2 = (\sqrt{85})^2
\]

Simplifying:

\[
(x - 5)^2 + (y - 3)^2 = 85
\]

So, the equation of the circle is:

\[
(x - 5)^2 + (y - 3)^2 = 85
\]

Would you like more details on this process or have any questions?

Here are 5 related questions for further exploration:

1. How would the equation change if the center was at the origin?
2. How can you find the area of the circle using the equation?
3. What is the general form of the equation of a circle?
4. How do you find the points where this circle intersects the x-axis or y-axis?
5. How can you determine if a given point lies inside, on, or outside this circle?

**Tip:** Always double-check the distance calculation when determining the radius, as any small mistake can lead to an incorrect equation!

Learn how to find the equation of a circle with center (5, 3) and passing through the point (-1, -4). This problem involves using the distance formula to determine the radius and then applying the equation of a circle formula. Suitable for students in Grades 10-12 studying coordinate geometry.

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