Find the equation of circle passing through the points (1,-2) and (3,-4) and touches the X-axis.

Answer

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

### Step 1: Understand the Circle's Properties
Since the circle touches the X-axis, the center of the circle must be at a point (h, k) where k is the radius of the circle. Therefore, the radius \( r \) is equal to \( |k| \).

### Step 2: Set Up the Circle's Equation
The general equation of a circle with center (h, k) and radius r is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Since the circle touches the X-axis, we have:
\[ r = |k| \]

### Step 3: Use the Circle's Properties
The points (1, -2) and (3, -4) lie on the circle, so they must satisfy the circle's equation. We can set up two equations based on these points.

1. For the point (1, -2):
\[ (1 - h)^2 + (-2 - k)^2 = k^2 \]
Expanding this gives:
\[ (1 - h)^2 + (k + 2)^2 = k^2 \]
\[ (1 - h)^2 + (k^2 + 4 + 4k) = k^2 \]
\[ (1 - h)^2 + 4 + 4k = 0 \quad \text{(Equation 1)} \]

2. For the point (3, -4):
\[ (3 - h)^2 + (-4 - k)^2 = k^2 \]
Expanding this gives:
\[ (3 - h)^2 + (k + 4)^2 = k^2 \]
\[ (3 - h)^2 + (k^2 + 16 + 8k) = k^2 \]
\[ (3 - h)^2 + 16 + 8k = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( (1 - h)^2 + 4 + 4k = 0 \)
2. \( (3 - h)^2 + 16 + 8k = 0 \)

We can simplify these equations:
1. From Equation 1:
\[ (1 - h)^2 + 4 + 4k = 0 \implies (1 - h)^2 = -4 - 4k \]
2. From Equation 2:
\[ (3 - h)^2 + 16 + 8k = 0 \implies (3 - h)^2 = -16 - 8k \]

### Step 5: Equate the Two Expressions
Now we can equate the two expressions derived from the equations:
\[ (1 - h)^2 + 4 + 4k = (3 - h)^2 + 16 + 8k \]
Expanding both sides:
\[ 1 - 2h + h^2 + 4 + 4k = 9 - 6h + h^2 + 16 + 8k \]
Simplifying gives:
\[ 5 - 2h + 4k = 25 - 6h + 8k \]
Rearranging terms:
\[ 4h - 4k = 20 \implies h - k = 5 \quad \text{(Equation 3)} \]

### Step 6: Substitute and Solve
Now substitute \( h = k + 5 \) into either Equation 1 or Equation 2. Let's use Equation 1:
\[ (1 - (k + 5))^2 + 4 + 4k = 0 \]
This simplifies to:
\[ (-4 - k)^2 + 4 + 4k = 0 \]
Expanding:
\[ 16 + 8k + k^2 + 4 + 4k = 0 \]
Combining like terms:
\[ k^2 + 12k + 20 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula:
\[ k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-12 \pm \sqrt{144 - 80}}{2} = \frac{-12 \pm \sqrt{64}}{2} = \frac{-12 \pm 8}{2} \]
Calculating gives:
\[ k = -2 \quad \text{or} \quad k = -10 \]

### Step 8: Find Corresponding h Values
For \( k = -2 \):
\[ h = -2 + 5 = 3 \]
For \( k = -10 \):
\[ h = -10 + 5 = -5 \]

### Step 9: Write the Circle Equations
1. For \( (h, k) = (3, -2) \):
\[ (x - 3)^2 + (y + 2)^2 = 4 \]
2. For \( (h, k) = (-5, -10) \):
\[ (x + 5)^2 + (y + 10)^2 = 100 \]

### Final Answer
The equations of the circles are:
1. \( (x - 3)^2 + (y + 2)^2 = 4 \)
2. \( (x + 5)^2 + (y + 10)^2 = 100 \)