Question

If the circles \( (x+1)^2 + (y+2)^2 = r^2 \) and \( x^2 + y^2 - 4x - 4y + 4 = 0 \) intersect at exactly two distinct points, then

• A. \( 3 < r < 7 \)
• B. \( 0 < r < 7 \)
• C. \( 5 < r < 9 \)
• D. \( \frac{1}{2} < r < 7 \)

Answer 1

The Correct Option is A

To find the range of r for which the circles intersect at exactly two points, we analyze the conditions for intersection.
The first circle has equation \((x + 1)^2 + (y + 2)^2 = r^2\), with center \(C_1 = (-1, -2)\) and radius \(r_1 = r\).
The second circle can be rewritten as \((x - 2)^2 + (y - 2)^2 = 9\), with center \(C_2 = (2, 2)\) and radius \(r_2 = 3\).
The distance \(d\) between \(C_1\) and \(C_2\) is:
\[ d = \sqrt{(2 - (-1))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
For two circles to intersect at exactly two points, the condition \(|r_1 - r_2| < d < r_1 + r_2\) must hold. Substitute \(r_1 = r\), \(r_2 = 3\), and \(d = 5\):
First inequality: \(|r - 3| < 5\)
Second inequality: \(5 < r + 3\)
Combining these results, we get:
\[ 3 < r < 7 \]

Answer 2

To determine the range of \( r \) for which the given circles intersect at exactly two distinct points, let's analyze the equations of both circles and use the concept of the distance between centers and their radii.

1. Identify the centers and radii of the circles:
   • First Circle: \((x+1)^2 + (y+2)^2 = r^2\)
     • Center: \((-1, -2)\)
     • Radius: \(r\)
   • Second Circle: \(x^2 + y^2 - 4x - 4y + 4 = 0\)
     • Rewrite it in standard form:
       \((x^2 - 4x) + (y^2 - 4y) = -4\)
     • Complete the square:
       \((x-2)^2 - 4 + (y-2)^2 - 4 = -4 \implies (x-2)^2 + (y-2)^2 = 4\)
     • Center: \((2, 2)\)
     • Radius: \(2\)
2. Calculate the distance \(d\) between the centers:
   • Distance formula: \(d = \sqrt{(2 - (-1))^2 + (2 - (-2))^2}\)
   • \(d = \sqrt{(3)^2 + (4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5\)
3. Determine the condition for two points of intersection:
   • The condition for the circles to intersect at exactly two points is:
     \(|r_1 - r_2| < d < r_1 + r_2\)
   • Here \( r_1 = r \) and \( r_2 = 2 \), and \( d = 5 \).
   • Therefore, the condition becomes: \(|r - 2| < 5 < r + 2\)
   • The first inequality: \(r - 2 < 5 \implies r < 7\)
   • The second inequality: \(r + 2 > 5 \implies r > 3\)
   • Combining these, \(3 < r < 7\)

Therefore, the correct answer is that the radius \( r \) should satisfy \(3 < r < 7\) for the circles to intersect at exactly two distinct points.