Key Concepts and Formulas

• Equation of a circle: The general equation of a circle is $x^2 + y^2 + 2gx + 2fy + c = 0$, where the center is $(-g, -f)$ and the radius is $r = \sqrt{g^2 + f^2 - c}$. The standard equation is $(x-h)^2 + (y-k)^2 = r^2$, with center $(h,k)$ and radius $r$.
• Distance between two points: The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Condition for circles touching internally: Two circles with centers $C_1$ and $C_2$, and radii $r_1$ and $r_2$ respectively, touch internally if the distance between their centers is equal to the absolute difference of their radii, i.e., $C_1C_2 = |r_1 - r_2|$. In this case, the equation of the common tangent is $S_1 - S_2 = 0$, where $S_1 = 0$ and $S_2 = 0$ are the equations of the circles.

Step-by-Step Solution

1. Find the Center and Radius of Circle 1

We are given the equation of the first circle as $x^2 + y^2 = 4$. This is in the standard form $(x-h)^2 + (y-k)^2 = r^2$.

• Comparing the given equation with the standard form, we can see that the center is $C_1 = (0, 0)$ and the radius is $r_1 = \sqrt{4} = 2$.
• This step is crucial to characterize the first circle fully, which is necessary to determine the relationship between the two circles.

2. Find the Center and Radius of Circle 2

The equation of the second circle is given as $x^2 + y^2 + 6x + 8y - 24 = 0$. This is in the general form $x^2 + y^2 + 2gx + 2fy + c = 0$.

• Comparing the given equation with the general form, we have $2g = 6$ and $2f = 8$, which gives $g = 3$ and $f = 4$. Therefore, the center is $C_2 = (-g, -f) = (-3, -4)$.
• The radius is given by $r_2 = \sqrt{g^2 + f^2 - c} = \sqrt{3^2 + 4^2 - (-24)} = \sqrt{9 + 16 + 24} = \sqrt{49} = 7$.
• Similar to Step 1, this step is crucial to characterize the second circle fully.

3. Determine the Relationship Between the Circles

We need to find the distance between the centers of the two circles and compare it with the sum and difference of their radii.

• The distance between the centers $C_1(0, 0)$ and $C_2(-3, -4)$ is given by: $C_1C_2 = \sqrt{(-3 - 0)^2 + (-4 - 0)^2} = \sqrt{(-3)^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

• The sum of the radii is $r_1 + r_2 = 2 + 7 = 9$.

• The absolute difference of the radii is $|r_1 - r_2| = |2 - 7| = |-5| = 5$.

• Since $C_1C_2 = |r_1 - r_2| = 5$, the two circles touch internally.

• Knowing the relationship between the circles is vital to choose the correct method for finding the common tangent.

4. Find the Equation of the Common Tangent

Since the circles touch internally, their common tangent is given by the equation $S_1 - S_2 = 0$, where $S_1 = x^2 + y^2 - 4 = 0$ and $S_2 = x^2 + y^2 + 6x + 8y - 24 = 0$.

• Substituting these into the formula: $(x^2 + y^2 - 4) - (x^2 + y^2 + 6x + 8y - 24) = 0$

• Simplifying the equation: $x^2 + y^2 - 4 - x^2 - y^2 - 6x - 8y + 24 = 0$ $-6x - 8y + 20 = 0$

• Dividing the equation by -2, we get: $3x + 4y - 10 = 0$

• This equation represents the common tangent to the two circles.

5. Check Which Point Lies on the Common Tangent

We need to substitute the coordinates of each given point into the equation of the common tangent $3x + 4y - 10 = 0$ and see which point satisfies the equation.

• (A) (6, -2): $3(6) + 4(-2) - 10 = 18 - 8 - 10 = 0$ This point satisfies the equation.

• (B) (4, -2): $3(4) + 4(-2) - 10 = 12 - 8 - 10 = -6 \neq 0$

• (C) (-4, 6): $3(-4) + 4(6) - 10 = -12 + 24 - 10 = 2 \neq 0$

• (D) (-6, 4): $3(-6) + 4(4) - 10 = -18 + 16 - 10 = -12 \neq 0$

Only the point (6, -2) satisfies the equation of the common tangent.

Common Mistakes & Tips

• Sign Errors: Pay close attention to signs, especially when calculating the radius and substituting values into equations. For example, in $r = \sqrt{g^2 + f^2 - c}$, ensure you correctly handle the sign of $c$.
• Center Confusion: Remember that the center of the circle $x^2 + y^2 + 2gx + 2fy + c = 0$ is $(-g, -f)$.
• Relationship of Circles: Correctly determine the relationship between the circles. If you incorrectly assume they intersect or touch externally, you will use the wrong method to find the common tangent.

Summary

We determined the centers and radii of the two circles, found that they touch internally, and then calculated the equation of the common tangent as $3x + 4y - 10 = 0$. By substituting the coordinates of the given points into the equation of the common tangent, we found that the point (6, -2) lies on the common tangent.

The final answer is \boxed{(6, –2)}, which corresponds to option (A).