Question

If two circles \( (x - 1)^2 + (y - 3)^2 = r^2 \) and \( x^2 + y^2 - 8x + 2y + 8 = 0 \) are given, then

• A. \( 2 < r < 8 \)
• B. \( r < 2 \)
• C. \( r = 2 \)
• D. \( r > 2 \)

Hint:

To find the relationship between two circles, simplify the equations and compare the centers and radii.

Answer 1

The Correct Option is A

We are given two equations of circles: 1. \( (x - 1)^2 + (y - 3)^2 = r^2 \), which represents a circle with center \( (1, 3) \). 2. \( x^2 + y^2 - 8x + 2y + 8 = 0 \), which we simplify to find the equation of the second circle. After simplifying the second circle's equation, we find the radius \( r \), and based on the relationship, the final result gives \( 2 < r < 8 \).