The equation of an ex - circle of a triangle formed by the common tangents to the circle `x^(2)+y^(2)=4 and x^(2)+y^(2)-6x+8=0` is

Options

A. `x^(2)+y^(2)=4`
B. `x^(2)+y^(2)-6x+8=0`
C. `x^(2)+y^(2)-6x+9=0`
D. `x^(2)+y^(2)=1`

Answer

To solve the problem of finding the equation of the ex-circle of a triangle formed by the common tangents to the circles given by the equations \(x^2 + y^2 = 4\) and \(x^2 + y^2 - 6x + 8 = 0\), we can follow these steps:

### Step 1: Identify the Circles
The first circle is given by the equation:
\[ x^2 + y^2 = 4 \]
This is a circle centered at \((0, 0)\) with a radius of \(2\).

The second circle can be rewritten by completing the square:
\[ x^2 + y^2 - 6x + 8 = 0 \implies (x - 3)^2 + y^2 = 1 \]
This shows that the second circle is centered at \((3, 0)\) with a radius of \(1\).

### Step 2: Determine the Positions of the Circles
- The first circle has center \((0, 0)\) and radius \(2\).
- The second circle has center \((3, 0)\) and radius \(1\).

Since the distance between the centers is \(3\) (which is greater than the sum of the radii \(2 + 1 = 3\)), the circles touch each other externally.

### Step 3: Identify the Ex-circle
In a triangle formed by the common tangents of these two circles, the larger circle will act as the ex-circle. The ex-circle is opposite to the vertex of the triangle formed by the tangents.

### Step 4: Write the Equation of the Ex-circle
The ex-circle will be centered at the point which is the external tangent point. The center of the larger circle is \((0, 0)\) and the center of the smaller circle is \((3, 0)\). The radius of the larger circle is \(2\).

The equation of the ex-circle can be expressed as:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.

For our case:
- Center \((h, k) = (0, 0)\)
- Radius \(r = 2\)

Thus, the equation of the ex-circle is:
\[ (x - 0)^2 + (y - 0)^2 = 2^2 \]
or simply:
\[ x^2 + y^2 = 4 \]

### Final Answer
The equation of the ex-circle of the triangle formed by the common tangents to the given circles is:
\[ \boxed{x^2 + y^2 = 4} \]