Question

Which of the following statement are false about two circles whose equations are given below? \[ x^2 + y^2 - 10x - 10y - 41 = 0 \] \[ x^2 + y^2 - 22x - 18y + 137 = 0 \]
• [A.] Circles have no common point
• [B.] Circles have one common point
• [C.] Circles have two common points
• [D.] Circles have infinitely many points

• A. B, C only
• B. A, D only
• C. B, D only
• D. A, C, D only

Hint:

Use distance between centers vs sum/difference of radii to determine intersection of circles.

Answer 1

The Correct Option is A

Concept: Distance between centers determines intersection.

Step 1: Convert to standard form. Circle 1: \[ (x-5)^2 + (y-5)^2 = 91 \] Circle 2: \[ (x-11)^2 + (y-9)^2 = 81 \]

Step 2: Find centers and radii. \[ C_1 = (5,5),\ r_1 = \sqrt{91} \] \[ C_2 = (11,9),\ r_2 = 9 \]

Step 3: Distance between centers. \[ d = \sqrt{(11-5)^2 + (9-5)^2} = \sqrt{36+16} = \sqrt{52} \]

Step 4: Compare values. \[ r_1 + r_2 > d \quad \text{and} \quad |r_1 - r_2| < d \] Thus circles intersect at two points.

Step 5: Conclusion.
• A false
• B false
• C true
• D false \[ \boxed{(1)} \]