Find the equation of the circle which passes through the points (3,4),(3,-6) and (1,2).

Answer

To find the equation of the circle that passes through the points (3,4), (3,-6), and (1,2), we can follow these steps:

### Step 1: General Equation of a Circle
The general equation of a circle can be expressed as:
\[ x^2 + y^2 + Dx + Ey + F = 0 \]
where \(D\), \(E\), and \(F\) are constants that we need to determine.

### Step 2: Substitute the Points into the Equation
We will substitute each of the three points into the general equation to create a system of equations.

1. For the point (3, 4):
\[ 3^2 + 4^2 + 3D + 4E + F = 0 \]
This simplifies to:
\[ 9 + 16 + 3D + 4E + F = 0 \quad \Rightarrow \quad 3D + 4E + F = -25 \quad \text{(Equation 1)} \]

2. For the point (3, -6):
\[ 3^2 + (-6)^2 + 3D - 6E + F = 0 \]
This simplifies to:
\[ 9 + 36 + 3D - 6E + F = 0 \quad \Rightarrow \quad 3D - 6E + F = -45 \quad \text{(Equation 2)} \]

3. For the point (1, 2):
\[ 1^2 + 2^2 + 1D + 2E + F = 0 \]
This simplifies to:
\[ 1 + 4 + D + 2E + F = 0 \quad \Rightarrow \quad D + 2E + F = -5 \quad \text{(Equation 3)} \]

### Step 3: Solve the System of Equations
Now we have the following system of equations:
1. \(3D + 4E + F = -25\) (Equation 1)
2. \(3D - 6E + F = -45\) (Equation 2)
3. \(D + 2E + F = -5\) (Equation 3)

We can solve these equations step by step.

#### Step 3.1: Eliminate \(F\)
Subtract Equation 3 from Equation 1:
\[ (3D + 4E + F) - (D + 2E + F) = -25 + 5 \]
This simplifies to:
\[ 2D + 2E = -20 \quad \Rightarrow \quad D + E = -10 \quad \text{(Equation 4)} \]

Subtract Equation 3 from Equation 2:
\[ (3D - 6E + F) - (D + 2E + F) = -45 + 5 \]
This simplifies to:
\[ 2D - 8E = -40 \quad \Rightarrow \quad D - 4E = -20 \quad \text{(Equation 5)} \]

#### Step 3.2: Solve Equations 4 and 5
Now we have:
1. \(D + E = -10\) (Equation 4)
2. \(D - 4E = -20\) (Equation 5)

From Equation 4, we can express \(D\) in terms of \(E\):
\[ D = -10 - E \]

Substituting \(D\) into Equation 5:
\[ (-10 - E) - 4E = -20 \]
This simplifies to:
\[ -10 - 5E = -20 \quad \Rightarrow \quad -5E = -10 \quad \Rightarrow \quad E = 2 \]

Substituting \(E = 2\) back into Equation 4:
\[ D + 2 = -10 \quad \Rightarrow \quad D = -12 \]

### Step 4: Substitute \(D\) and \(E\) into Equation 3
Now we can find \(F\) using Equation 3:
\[ D + 2E + F = -5 \]
Substituting \(D = -12\) and \(E = 2\):
\[ -12 + 4 + F = -5 \quad \Rightarrow \quad F = 3 \]

### Step 5: Write the Final Equation of the Circle
Now we have \(D = -12\), \(E = 2\), and \(F = 3\). The equation of the circle is:
\[ x^2 + y^2 - 12x + 2y + 3 = 0 \]

### Step 6: Rearranging the Equation
Rearranging gives us:
\[ x^2 + y^2 - 12x + 2y + 3 = 0 \]

### Final Answer
The equation of the circle is:
\[ x^2 + y^2 - 12x + 2y + 3 = 0 \]

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