Find the equation of a circle (2,3),(3,4),(-1,2)

To find the equation of a circle passing through the points \((2, 3)\), \((3, 4)\), and \((-1, 2)\), we can use the general equation of a circle:

\[
x^2 + y^2 + Dx + Ey + F = 0
\]

where \(D\), \(E\), and \(F\) are constants we need to determine.

### Step 1: Set up the system of equations
Substitute the three points into the general equation of the circle:

For \((2, 3)\):
\[
(2)^2 + (3)^2 + 2D + 3E + F = 0 \quad \Rightarrow \quad 4 + 9 + 2D + 3E + F = 0 \quad \Rightarrow \quad 2D + 3E + F = -13
\]

For \((3, 4)\):
\[
(3)^2 + (4)^2 + 3D + 4E + F = 0 \quad \Rightarrow \quad 9 + 16 + 3D + 4E + F = 0 \quad \Rightarrow \quad 3D + 4E + F = -25
\]

For \((-1, 2)\):
\[
(-1)^2 + (2)^2 + (-1)D + 2E + F = 0 \quad \Rightarrow \quad 1 + 4 - D + 2E + F = 0 \quad \Rightarrow \quad -D + 2E + F = -5
\]

### Step 2: Solve the system of equations
Now we have the following system of equations:

\[
\begin{aligned}
1) & \quad 2D + 3E + F = -13 \\
2) & \quad 3D + 4E + F = -25 \\
3) & \quad -D + 2E + F = -5
\end{aligned}
\]

Let's solve this system step by step. We can solve it using elimination or substitution.

I'll solve it now.

The solution to the system of equations is:

\[
D = 2, \quad E = -14, \quad F = 25
\]

### Step 3: Write the equation of the circle

Substitute \(D = 2\), \(E = -14\), and \(F = 25\) into the general equation of the circle:

\[
x^2 + y^2 + 2x - 14y + 25 = 0
\]

Thus, the equation of the circle passing through the points \((2, 3)\), \((3, 4)\), and \((-1, 2)\) is:

\[
x^2 + y^2 + 2x - 14y + 25 = 0
\]

Would you like further details on any step?

Here are some related questions you might find interesting:
1. How do you derive the equation of a circle from its center and radius?
2. Can a circle pass through more than three points?
3. What are the properties of the general equation of a circle?
4. How can we verify if a point lies on a given circle?
5. What happens if the three points are collinear when finding a circle?

**Tip:** You can check if the points are collinear by ensuring the determinant of a matrix with these points is non-zero.

Find the equation of a circle passing through the points (2,3), (3,4), (-1,2)

Learn how to find the equation of a circle passing through the points (2,3), (3,4), and (-1,2). This problem involves using the general equation of a circle and solving a system of equations step by step. Suitable for high school students in Grades 10-12.

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