The common tangent to the circles `x^(2)+y^(2) = 4 and x^(2) + y^(2) + 6x + 8y - 24 = 0` also passes through the point

Options

A. (4, -2)
B. (-4, 6)
C. (-6, 4)
D. (6, -2)

Explanation

To solve the problem of finding the common tangent to the circles given by the equations \(x^2 + y^2 = 4\) and \(x^2 + y^2 + 6x + 8y - 24 = 0\), and to check which point among the given options lies on this tangent, we can follow these steps:

### Step 1: Identify the circles
The first circle is given by:
\[ x^2 + y^2 = 4 \]
This circle has a center at \((0, 0)\) and a radius \(r_1 = 2\).

The second circle can be rewritten by completing the square:
\[ x^2 + y^2 + 6x + 8y - 24 = 0 \]
Rearranging gives:
\[ (x^2 + 6x) + (y^2 + 8y) = 24 \]
Completing the square for \(x\) and \(y\):
\[ (x + 3)^2 - 9 + (y + 4)^2 - 16 = 24 \]
This simplifies to:
\[ (x + 3)^2 + (y + 4)^2 = 49 \]
Thus, the second circle has a center at \((-3, -4)\) and a radius \(r_2 = 7\).

### Step 2: Find the equation of the common tangent
The equation of the common tangent can be derived using the formula for the tangent between two circles:
\[ S_1 - S_2 = 0 \]
where \(S_1\) and \(S_2\) are the equations of the circles.

Substituting the equations:
\[ (x^2 + y^2 - 4) - (x^2 + y^2 + 6x + 8y - 24) = 0 \]
This simplifies to:
\[ -6x - 8y + 20 = 0 \]
Rearranging gives:
\[ 6x + 8y - 20 = 0 \]
Dividing the entire equation by 2:
\[ 3x + 4y - 10 = 0 \]

### Step 3: Check which point lies on the tangent
We need to check which of the given points satisfies the equation \(3x + 4y - 10 = 0\).

1. **For the point (4, -2)**:
\[ 3(4) + 4(-2) - 10 = 12 - 8 - 10 = -6 \quad (\text{not on the tangent}) \]

2. **For the point (-4, 6)**:
\[ 3(-4) + 4(6) - 10 = -12 + 24 - 10 = 2 \quad (\text{not on the tangent}) \]

3. **For the point (-6, 4)**:
\[ 3(-6) + 4(4) - 10 = -18 + 16 - 10 = -12 \quad (\text{not on the tangent}) \]

4. **For the point (6, -2)**:
\[ 3(6) + 4(-2) - 10 = 18 - 8 - 10 = 0 \quad (\text{on the tangent}) \]

### Conclusion
The common tangent to the circles passes through the point \((6, -2)\).