The circles `x^(2)+y^(2)-4x+6y+8=0` and `x^(2)+y^(2)-10x-6y+14=0`

Options

A. touch externally
B. touch internally
C. intersect
D. do not touch

Answer

To determine the relationship between the two circles given by the equations \(x^2 + y^2 - 4x + 6y + 8 = 0\) and \(x^2 + y^2 - 10x - 6y + 14 = 0\), we will follow these steps:

### Step 1: Rewrite the equations in standard form
We start by rewriting each circle's equation in the standard form \((x - h)^2 + (y - k)^2 = r^2\).

**For Circle 1:**
Given equation:
\[ x^2 + y^2 - 4x + 6y + 8 = 0 \]
Rearranging gives:
\[ x^2 - 4x + y^2 + 6y = -8 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 - 4x + 4) + (y^2 + 6y + 9) = -8 + 4 + 9 \]
\[ (x - 2)^2 + (y + 3)^2 = 5 \]
Thus, the center \(C_1\) is \((2, -3)\) and the radius \(r_1 = \sqrt{5}\).

**For Circle 2:**
Given equation:
\[ x^2 + y^2 - 10x - 6y + 14 = 0 \]
Rearranging gives:
\[ x^2 - 10x + y^2 - 6y = -14 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 - 10x + 25) + (y^2 - 6y + 9) = -14 + 25 + 9 \]
\[ (x - 5)^2 + (y - 3)^2 = 20 \]
Thus, the center \(C_2\) is \((5, 3)\) and the radius \(r_2 = \sqrt{20} = 2\sqrt{5}\).

### Step 2: Calculate the distance between the centers
The distance \(d\) between the centers \(C_1(2, -3)\) and \(C_2(5, 3)\) is given by:
\[ d = \sqrt{(5 - 2)^2 + (3 + 3)^2} = \sqrt{3^2 + 6^2} = \sqrt{9 + 36} = \sqrt{45} = 3\sqrt{5} \]

### Step 3: Check the conditions for touching or intersecting
The circles can either touch externally, touch internally, or intersect based on the following conditions:

1. **Touch externally**: \(d = r_1 + r_2\)
2. **Touch internally**: \(d = |r_1 - r_2|\)
3. **Intersect**: \( |r_1 - r_2| < d < r_1 + r_2\)

Calculating \(r_1 + r_2\) and \(|r_1 - r_2|\):
\[ r_1 + r_2 = \sqrt{5} + 2\sqrt{5} = 3\sqrt{5} \]
\[ |r_1 - r_2| = | \sqrt{5} - 2\sqrt{5} | = | -\sqrt{5} | = \sqrt{5} \]

### Step 4: Compare the distance with the radii
Now we compare:
- \(d = 3\sqrt{5}\)
- \(r_1 + r_2 = 3\sqrt{5}\)
- \(|r_1 - r_2| = \sqrt{5}\)

Since \(d = r_1 + r_2\), the circles touch externally.

### Final Conclusion
The two circles touch externally.

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