The equation of a circle touching x-axis at `(-4, 0)` and cutting off an intercept of 6 units on y-axis can be

Options

A. `x^(2) + y^(2) + 8x + 12y + 16 = 0`
B. `x^(2) + y^(2) + 8x - 12y + 16 = 0`
C. `x^(2) + y^(2) + 8x + 10 y + 16 = 0`
D. `x^(2) + y^(2) + 8x - 10y + 16 = 0`

Answer

To find the equation of the circle that touches the x-axis at the point (-4, 0) and cuts off an intercept of 6 units on the y-axis, we can follow these steps:

### Step 1: Understand the properties of the circle
The circle touches the x-axis at (-4, 0), which means that the center of the circle is directly above or below this point. Since the circle touches the x-axis, the radius of the circle is equal to the y-coordinate of the center.

### Step 2: Define the center of the circle
Let the center of the circle be at the point (-4, h), where h is the y-coordinate of the center. The radius of the circle (r) is then equal to |h|.

### Step 3: Determine the y-intercept
The problem states that the circle cuts off a y-intercept of 6 units. This means that the distance from the center of the circle to the y-axis is 6 units. The y-intercept can be found using the formula for the distance from the center to the y-axis, which is given by the absolute value of the x-coordinate of the center.

### Step 4: Set up the equation for the y-intercept
Since the center is at (-4, h), the distance to the y-axis is |-4| = 4. The total distance from the center to the y-axis must equal 6 units. Therefore, we can write the equation:
\[ |h| + 4 = 6 \]

### Step 5: Solve for h
From the equation above, we can solve for h:
\[ |h| = 6 - 4 = 2 \]
This gives us two possible values for h:
1. \( h = 2 \)
2. \( h = -2 \)

### Step 6: Write the equations of the circles
Now we can write the equations of the circles for both cases.

1. **For h = 2** (circle above the x-axis):
The center is (-4, 2) and the radius is 2.
The equation of the circle is:
\[ (x + 4)^2 + (y - 2)^2 = 2^2 \]
Simplifying gives:
\[ (x + 4)^2 + (y - 2)^2 = 4 \]

2. **For h = -2** (circle below the x-axis):
The center is (-4, -2) and the radius is 2.
The equation of the circle is:
\[ (x + 4)^2 + (y + 2)^2 = 2^2 \]
Simplifying gives:
\[ (x + 4)^2 + (y + 2)^2 = 4 \]

### Step 7: Final equations
Thus, the equations of the circles that satisfy the given conditions are:
1. \( (x + 4)^2 + (y - 2)^2 = 4 \)
2. \( (x + 4)^2 + (y + 2)^2 = 4 \)