Two circles `x^2 + y^2 + 2x-4y=0 and x^2 + y^2 - 8y - 4 = 0` (A) touch each other externally (B) intersect each other (C) touch each other internally (D) none of these

Answer

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

### Step 1: Rewrite the equations of the circles in standard form
The general form of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From this, we can extract the center and radius.

**Circle 1:**
\[ x^2 + y^2 + 2x - 4y = 0 \]
Rearranging gives:
\[ x^2 + y^2 + 2x - 4y = 0 \]
This can be rewritten as:
\[ (x^2 + 2x) + (y^2 - 4y) = 0 \]
Completing the square:
\[ (x + 1)^2 - 1 + (y - 2)^2 - 4 = 0 \]
\[ (x + 1)^2 + (y - 2)^2 = 5 \]
Thus, the center \(C_1\) is \((-1, 2)\) and the radius \(r_1\) is \(\sqrt{5}\).

**Circle 2:**
\[ x^2 + y^2 - 8y - 4 = 0 \]
Rearranging gives:
\[ x^2 + (y^2 - 8y) = 4 \]
Completing the square:
\[ x^2 + (y - 4)^2 - 16 = 4 \]
\[ x^2 + (y - 4)^2 = 20 \]
Thus, the center \(C_2\) is \((0, 4)\) and the radius \(r_2\) is \(2\sqrt{5}\).

### Step 2: Calculate the distance between the centers
Using the distance formula, the distance \(d\) between the centers \(C_1\) and \(C_2\) is:
\[ d = \sqrt{(0 - (-1))^2 + (4 - 2)^2} = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 3: Determine the relationship between the circles
To determine if the circles touch externally, touch internally, or intersect, we compare the distance between the centers \(d\) with the sum and the difference of the radii.

- The sum of the radii:
\[ r_1 + r_2 = \sqrt{5} + 2\sqrt{5} = 3\sqrt{5} \]

- The difference of the radii:
\[ |r_2 - r_1| = |2\sqrt{5} - \sqrt{5}| = |(2 - 1)\sqrt{5}| = \sqrt{5} \]

### Step 4: Analyze the conditions
- If \(d = r_1 + r_2\), the circles touch externally.
- If \(d = |r_2 - r_1|\), the circles touch internally.
- If \(d < r_1 + r_2\) and \(d > |r_2 - r_1|\), the circles intersect.
- If \(d > r_1 + r_2\), the circles are separate.

In our case:
- \(d = \sqrt{5}\)
- \(r_2 - r_1 = \sqrt{5}\)

Since \(d = |r_2 - r_1|\), the circles touch each other internally.

### Conclusion
The circles touch each other internally. The correct option is (C).

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