Key Concepts and Formulas

• Midpoint Formula: The midpoint of a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$ is given by $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$.
• Distance Between Parallel Lines: The distance between two parallel lines $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$ is given by $\frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}$.
• Equation of a Circle: The equation of a circle with center $(h, k)$ and radius $r$ is $(x - h)^2 + (y - k)^2 = r^2$.
• Circle Tangent to Parallel Lines: The center of a circle tangent to two parallel lines lies on the line midway between them, and the diameter of the circle equals the distance between the lines.

Step-by-Step Solution

Step 1: Find the center of the circle.

• What we are doing: We are given that the center of the circle lies on the line connecting the intersection points of a line through the center with the two tangent lines. The center is equidistant from both lines. Thus, the center is the midpoint of the segment connecting the two intersection points.
• Why: Since the center of the circle lies on the line passing through $(-1, 2)$ and $(3, -6)$, and is equidistant from the two parallel lines, it must be the midpoint of the segment joining these two points.
• Calculation: Using the midpoint formula with $(-1, 2)$ and $(3, -6)$, we find the center $(h, k)$: $h = \frac{-1 + 3}{2} = \frac{2}{2} = 1$ $k = \frac{2 + (-6)}{2} = \frac{-4}{2} = -2$ Thus, the center of the circle is $(1, -2)$.

Step 2: Determine the equations of the tangent lines.

• What we are doing: We will substitute the given points into the equations of the respective lines to find $K_1$ and $K_2$.
• Why: We need to find the exact equations of the parallel lines to calculate the distance between them, which will give us the diameter of the circle. We use the given intersection points to find $K_1$ and $K_2$.
• Calculation:
  • Line $L_1$: $4x - 3y + K_1 = 0$ passes through $(-1, 2)$. Substituting these coordinates: $4(-1) - 3(2) + K_1 = 0$ $-4 - 6 + K_1 = 0$ $K_1 = 10$ So, $L_1$ is $4x - 3y + 10 = 0$.
  • Line $L_2$: $4x - 3y + K_2 = 0$ passes through $(3, -6)$. Substituting these coordinates: $4(3) - 3(-6) + K_2 = 0$ $12 + 18 + K_2 = 0$ $K_2 = -30$ So, $L_2$ is $4x - 3y - 30 = 0$.

Step 3: Calculate the radius of the circle.

• What we are doing: We will calculate the distance between the two parallel lines to find the diameter of the circle, then divide by 2 to find the radius.
• Why: The distance between the two tangent lines is equal to the diameter of the circle.
• Calculation: The distance $D$ between the parallel lines $4x - 3y + 10 = 0$ and $4x - 3y - 30 = 0$ is: $D = \frac{|10 - (-30)|}{\sqrt{4^2 + (-3)^2}} = \frac{|40|}{\sqrt{16 + 9}} = \frac{40}{\sqrt{25}} = \frac{40}{5} = 8$ The radius $r$ is half the diameter: $r = \frac{D}{2} = \frac{8}{2} = 4$

Step 4: Write the equation of the circle.

• What we are doing: We are going to use the center and radius to write the equation of the circle in standard form.
• Why: We have all the information needed to construct the equation of the circle.
• Calculation: The equation of the circle with center $(1, -2)$ and radius $4$ is: $(x - 1)^2 + (y - (-2))^2 = 4^2$ $(x - 1)^2 + (y + 2)^2 = 16$

Common Mistakes & Tips

• Be careful with signs when substituting the coordinates of the center into the equation of the circle.
• Remember to use the radius, not the diameter, in the equation of the circle.
• Double-check your arithmetic, especially when calculating the midpoint and the distance between the parallel lines.

Summary

We found the center of the circle by using the midpoint formula. Then, we used the coordinates of the points on the tangent lines to determine the values of $K_1$ and $K_2$. With the equations of the lines known, we calculated the distance between them, which gave us the diameter of the circle. Finally, we used the center and the radius to write the equation of the circle in standard form.

The final answer is \boxed{(x - 1)^2 + (y + 2)^2 = 16}, which corresponds to option (C).