The two circles `x^(2)+y^(2)-5=0` and `x^(2)+y^(2)-2x-4y-15=0`

Options

A. touch each other externally
B. touch each other internally
C. cut each other orthogonally
D. do not intersect

Answer

To solve the problem involving the two circles given by the equations \(x^2 + y^2 - 5 = 0\) and \(x^2 + y^2 - 2x - 4y - 15 = 0\), we will follow these steps:

### Step 1: Identify the first circle's center and radius.
The first circle's equation is:
\[ x^2 + y^2 - 5 = 0 \]
This can be rewritten as:
\[ x^2 + y^2 = 5 \]
From this equation, we can identify the center \((h, k)\) and the radius \(r\):
- Center: \((0, 0)\)
- Radius: \(r_1 = \sqrt{5}\)

### Step 2: Identify the second circle's center and radius.
The second circle's equation is:
\[ x^2 + y^2 - 2x - 4y - 15 = 0 \]
We can rearrange this equation by completing the square:
1. Rearranging gives:
\[ x^2 - 2x + y^2 - 4y = 15 \]
2. Completing the square for \(x\):
\[ (x - 1)^2 - 1 \]
3. Completing the square for \(y\):
\[ (y - 2)^2 - 4 \]
4. Substitute back:
\[ (x - 1)^2 - 1 + (y - 2)^2 - 4 = 15 \]
Simplifying gives:
\[ (x - 1)^2 + (y - 2)^2 = 20 \]
From this equation, we can identify the center and radius:
- Center: \((1, 2)\)
- Radius: \(r_2 = \sqrt{20} = 2\sqrt{5}\)

### Step 3: Calculate the distance between the centers.
The distance \(d\) between the centers \((0, 0)\) and \((1, 2)\) is calculated using the distance formula:
\[ d = \sqrt{(1 - 0)^2 + (2 - 0)^2} = \sqrt{1^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 4: Compare the distance with the sum and difference of the radii.
Now, we calculate the sum and difference of the radii:
- Sum of the radii:
\[ r_1 + r_2 = \sqrt{5} + 2\sqrt{5} = 3\sqrt{5} \]
- Difference of the radii:
\[ r_2 - r_1 = 2\sqrt{5} - \sqrt{5} = \sqrt{5} \]

### Step 5: Determine the relationship between the circles.
We compare the distance \(d\) with the sum and difference of the radii:
- Since \(d = \sqrt{5}\) (the distance between centers) is equal to \(r_2 - r_1 = \sqrt{5}\) (the difference of the radii), this means that the circles touch internally.

### Conclusion:
The two circles touch internally.

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