Find the equation of the circle passing through the point: (1,2),(3,-4) and (5,-6).

Options

A. `9`
B. `5`
C. `1.5`
D. `0`

Explanation

To find the equation of the circle passing through the points (1, 2), (3, -4), and (5, -6), we can follow these steps:

### Step 1: Write the general equation of the 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 given points into the equation to create a system of equations.

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

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

3. For the point (5, -6):
\[ 5^2 + (-6)^2 + D(5) + E(-6) + F = 0 \implies 25 + 36 + 5D - 6E + F = 0 \implies 5D - 6E + F = -61 \quad \text{(Equation 3)} \]

### Step 3: Solve the system of equations
Now we have a system of three equations:
1. \(D + 2E + F = -5\) (Equation 1)
2. \(3D - 4E + F = -25\) (Equation 2)
3. \(5D - 6E + F = -61\) (Equation 3)

We can eliminate \(F\) by subtracting Equation 1 from Equations 2 and 3.

**Subtract Equation 1 from Equation 2:**
\[ (3D - 4E + F) - (D + 2E + F) = -25 + 5 \]
\[ 2D - 6E = -20 \implies D - 3E = -10 \quad \text{(Equation 4)} \]

**Subtract Equation 1 from Equation 3:**
\[ (5D - 6E + F) - (D + 2E + F) = -61 + 5 \]
\[ 4D - 8E = -56 \implies D - 2E = -14 \quad \text{(Equation 5)} \]

### Step 4: Solve Equations 4 and 5
Now we have:
1. \(D - 3E = -10\) (Equation 4)
2. \(D - 2E = -14\) (Equation 5)

Subtract Equation 5 from Equation 4:
\[ (D - 3E) - (D - 2E) = -10 + 14 \]
\[ -E = 4 \implies E = -4 \]

Substituting \(E = -4\) into Equation 5:
\[ D - 2(-4) = -14 \implies D + 8 = -14 \implies D = -22 \]

### Step 5: Find \(F\)
Now substitute \(D\) and \(E\) back into Equation 1 to find \(F\):
\[ -22 + 2(-4) + F = -5 \]
\[ -22 - 8 + F = -5 \implies F = 25 \]

### Step 6: Write the final equation of the circle
Now we have \(D = -22\), \(E = -4\), and \(F = 25\). The equation of the circle is:
\[ x^2 + y^2 - 22x - 4y + 25 = 0 \]

### Step 7: Simplify (if necessary)
This can be rearranged to:
\[ x^2 + y^2 - 22x - 4y + 25 = 0 \]