The number of common tangents to the circles `x^(2)+y^(2)-4x-2y+k` =0 and `x^(2)+y^(2)-6x-4y+l`=0 having radii 2 and 3 respectively is

Answer

To find the number of common tangents to the circles given by the equations \(x^2 + y^2 - 4x - 2y + k = 0\) and \(x^2 + y^2 - 6x - 4y + l = 0\), we will follow these steps:

### Step 1: Identify the centers and radii of the circles

The general form of a circle is given by the equation:
\[ x^2 + y^2 - 2gx - 2fy + c = 0 \]
From this, we can identify the center \((g, f)\) and the radius \(r\) of the circle.

1. For the first circle \(x^2 + y^2 - 4x - 2y + k = 0\):
- Here, \(g = 2\) and \(f = 1\).
- The radius \(r_1\) is given as \(2\).

2. For the second circle \(x^2 + y^2 - 6x - 4y + l = 0\):
- Here, \(g = 3\) and \(f = 2\).
- The radius \(r_2\) is given as \(3\).

### Step 2: Calculate the distance between the centers of the circles

The centers of the circles are:
- Center of Circle 1: \(C_1(2, 1)\)
- Center of Circle 2: \(C_2(3, 2)\)

The distance \(d\) between the centers \(C_1\) and \(C_2\) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates:
\[ d = \sqrt{(3 - 2)^2 + (2 - 1)^2} = \sqrt{1^2 + 1^2} = \sqrt{2} \]

### Step 3: Analyze the relationship between the distance and the radii

Now we compare the distance \(d\) with the sum and difference of the radii:
- Sum of the radii: \(r_1 + r_2 = 2 + 3 = 5\)
- Difference of the radii: \(|r_1 - r_2| = |2 - 3| = 1\)

### Step 4: Determine the number of common tangents

We have:
- \(d = \sqrt{2} \approx 1.41\)
- \(r_1 + r_2 = 5\)
- \(|r_1 - r_2| = 1\)

Since \(d < r_1 + r_2\) and \(d > |r_1 - r_2|\), the circles intersect at two points. When two circles intersect, there are exactly two common tangents.

### Conclusion

Thus, the number of common tangents to the given circles is **2**.

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