Question

The point \( (-3, -2) \) lies

• A. inside \( S' \) only
• B. inside \( S \) only
• C. inside \( S \) and \( S' \)
• D. outside \( S \) and \( S' \)

Hint:

When determining whether a point lies inside or outside a circle, use the distance formula to calculate the distance from the point to the center of the circle and compare it with the radius.

Answer 1

The Correct Option is A

We are given two equations of circles: 1. \( S: (x - 1)^2 + (y - 3)^2 = r^2 \) 2. \( S': x^2 + y^2 - 8x + 2y + 8 = 0 \) For the second circle, we complete the square: \[ x^2 - 8x + y^2 + 2y = -8 \quad \Rightarrow \quad (x - 4)^2 + (y + 1)^2 = 7 \] Thus, the center of \( S' \) is \( (4, -1) \) and the radius is \( \sqrt{7} \). Now, for the first circle \( S \), the center is \( (1, 3) \) and the radius is \( r \). Next, calculate the distance from the point \( (-3, -2) \) to the centers of both circles: 1. Distance from \( (-3, -2) \) to \( (1, 3) \): \[ \text{Distance} = \sqrt{(-3 - 1)^2 + (-2 - 3)^2} = \sqrt{16 + 25} = \sqrt{41} \] Since \( \sqrt{41} < r \), the point lies outside \( S \). 2. Distance from \( (-3, -2) \) to \( (4, -1) \): \[ \text{Distance} = \sqrt{(-3 - 4)^2 + (-2 + 1)^2} = \sqrt{49 + 1} = \sqrt{50} \] Since \( \sqrt{50} < \sqrt{7} \), the point lies inside \( S' \). Thus, the point \( (-3, -2) \) lies inside \( S' \) only.