To determine the equation of a circle, we need to use the general form for a circle centered at $(h, k)$ with radius $r$:

$(x - h)^2 + (y - k)^2 = r^2$

Given Information

• Center of the circle: $(h, k) = (2, -3)$
• The circle passes through the point $(4, 1)$.

Step-by-Step Solution

1. Set up the equation with the center: Substituting $h = 2$ and $k = -3$ into the general equation, we get: $(x - 2)^2 + (y + 3)^2 = r^2$

2. Calculate the radius: Since the circle passes through the point $(4, 1)$, we can use this point to find $r$.

   Substitute $(x, y) = (4, 1)$ into the equation: $(4 - 2)^2 + (1 + 3)^2 = r^2$ Simplifying: $(2)^2 + (4)^2 = r^2$ $4 + 16 = r^2$ $r^2 = 20$

3. Write the final equation: Substitute $r^2 = 20$ back into the circle equation: $(x - 2)^2 + (y + 3)^2 = 20$

Final Answer

The equation of the circle is: $(x - 2)^2 + (y + 3)^2 = 20$

Would you like more details on any specific part of this solution, or do you have further questions?

Related Questions

1. How can we find the radius of a circle given two points on the circumference?
2. What is the distance formula used for in geometry, and how is it applied to circles?
3. How can you determine if a given point lies on, inside, or outside a circle?
4. What other forms can the equation of a circle take, and when are they used?
5. How do you convert the general form of a circle equation to the standard form?

Tip

When finding the equation of a circle given two points, always check if one point is the center to simplify the radius calculation using the distance formula.