Two circles ` x^(2) + y^(2) - 4x + 10y + 20 = 0 ` and
` x^(2) + y^(2) + 8x - 6y - 24= 0`

Options

A. touch externally
B. touch internally
C. are orthogonal
D. are disjoint

Answer

To find the relationship between the two circles given by the equations \( x^2 + y^2 - 4x + 10y + 20 = 0 \) and \( x^2 + y^2 + 8x - 6y - 24 = 0 \), we can follow these steps:

### Step 1: Rewrite the equations in standard form
The standard form of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
where \((h, k)\) is the center and \(r\) is the radius.

**For the first circle:**
\[ x^2 + y^2 - 4x + 10y + 20 = 0 \]
Rearranging gives:
\[ x^2 - 4x + y^2 + 10y + 20 = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 - 4x + 4) + (y^2 + 10y + 25) = -20 + 4 + 25 \]
This simplifies to:
\[ (x - 2)^2 + (y + 5)^2 = 1 \]
Thus, the center \(C_1\) is \((2, -5)\) and the radius \(R_1 = 1\).

**For the second circle:**
\[ x^2 + y^2 + 8x - 6y - 24 = 0 \]
Rearranging gives:
\[ x^2 + 8x + y^2 - 6y - 24 = 0 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 + 8x + 16) + (y^2 - 6y + 9) = 24 + 16 + 9 \]
This simplifies to:
\[ (x + 4)^2 + (y - 3)^2 = 49 \]
Thus, the center \(C_2\) is \((-4, 3)\) and the radius \(R_2 = 7\).

### Step 2: Calculate the distance between the centers
The distance \(d\) between the centers \(C_1(2, -5)\) and \(C_2(-4, 3)\) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{((-4) - 2)^2 + (3 - (-5))^2} = \sqrt{(-6)^2 + (8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 3: Determine the relationship between the circles
We compare the distance \(d\) with the sum of the radii \(R_1 + R_2\):
\[ R_1 + R_2 = 1 + 7 = 8 \]
Since \(d = 10\) and \(R_1 + R_2 = 8\), we find that:
\[ d > R_1 + R_2 \]
This means the circles do not touch or intersect; they are separate.

### Conclusion
The two circles are separate and do not intersect.

---