Discuss the relative position of the following pair of circles.
` (x-2)^(2) + (y+ 1) ^(2) = 9, (x+ 1) ^(2) + (y-3)^(2) = 4`

Answer

To discuss the relative position of the given pair of circles, we will follow these steps:

### Step 1: Write the equations of the circles in standard form

The equations of the circles are:
1. \((x - 2)^2 + (y + 1)^2 = 9\)
2. \((x + 1)^2 + (y - 3)^2 = 4\)

### Step 2: Identify the center and radius of each circle

For the first circle:
- The center \((h, k)\) can be derived from the equation \((x - h)^2 + (y - k)^2 = r^2\).
- Here, \(h = 2\), \(k = -1\), and \(r^2 = 9\) which gives \(r = 3\).

Thus, the center of the first circle is \(C_1(2, -1)\) and the radius \(r_1 = 3\).

For the second circle:
- Here, \(h = -1\), \(k = 3\), and \(r^2 = 4\) which gives \(r = 2\).

Thus, the center of the second circle is \(C_2(-1, 3)\) and the radius \(r_2 = 2\).

### Step 3: Calculate the distance between the centers of the circles

To find the distance \(D\) between the centers \(C_1(2, -1)\) and \(C_2(-1, 3)\), we use the distance formula:
\[ D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ D = \sqrt{((-1) - 2)^2 + (3 - (-1))^2} = \sqrt{(-3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

### Step 4: Compare the distance with the sum of the radii

Now, we calculate the sum of the radii:
\[ r_1 + r_2 = 3 + 2 = 5 \]

### Step 5: Determine the relative position of the circles

Since the distance \(D\) between the centers \(5\) is equal to the sum of the radii \(r_1 + r_2 = 5\), we conclude that the two circles touch each other externally.

### Final Conclusion

The two circles touch each other externally.

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