The circles `x^(2)+y^(2)-8x+6y+21=0, x^(2)+y^(2)+4x-10y-115=0` are a) intersecting b) touching externally c) touching internally d) one is lying inside the other

Options

A. intersecting
B. touching externally
C. touching internally
D. one is lying inside the other

Answer

To determine the relationship between the two circles given by the equations \(x^2 + y^2 - 8x + 6y + 21 = 0\) and \(x^2 + y^2 + 4x - 10y - 115 = 0\), we will follow these steps:

### Step 1: Rewrite the equations in standard form
We need to convert both equations into the standard form of a circle, which is \((x - h)^2 + (y - k)^2 = r^2\).

#### For the first circle:
Starting with the equation:
\[ x^2 + y^2 - 8x + 6y + 21 = 0 \]
Rearranging gives:
\[ x^2 - 8x + y^2 + 6y = -21 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 - 8x + 16) + (y^2 + 6y + 9) = -21 + 16 + 9 \]
This simplifies to:
\[ (x - 4)^2 + (y + 3)^2 = 4 \]
Thus, the center \(C_1\) is \((4, -3)\) and the radius \(r_1 = 2\).

#### For the second circle:
Starting with the equation:
\[ x^2 + y^2 + 4x - 10y - 115 = 0 \]
Rearranging gives:
\[ x^2 + 4x + y^2 - 10y = 115 \]
Completing the square for \(x\) and \(y\):
\[ (x^2 + 4x + 4) + (y^2 - 10y + 25) = 115 + 4 + 25 \]
This simplifies to:
\[ (x + 2)^2 + (y - 5)^2 = 144 \]
Thus, the center \(C_2\) is \((-2, 5)\) and the radius \(r_2 = 12\).

### Step 2: Calculate the distance between the centers
Now we calculate the distance \(d\) between the centers \(C_1\) and \(C_2\):
\[ d = \sqrt{(4 - (-2))^2 + ((-3) - 5)^2} \]
Calculating the components:
\[ d = \sqrt{(4 + 2)^2 + (-3 - 5)^2} = \sqrt{6^2 + (-8)^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]

### Step 3: Compare the distance with the radii
Now we compare the distance \(d\) with the sum and difference of the radii:
- \(r_1 = 2\)
- \(r_2 = 12\)

Calculating the difference of the radii:
\[ r_2 - r_1 = 12 - 2 = 10 \]

### Step 4: Determine the relationship
Since \(d = r_2 - r_1\), we conclude that the circles touch internally.

### Final Answer
The correct option is:
**c) touching internally**