Q.9 A circle with center at (x, y) = (0.5, 0) and radius = 0.5 intersects with another circle with center at (x,
y) = (1, 1) and radius = 1 at two points. One of the points of intersection (x,y) is:
(A)(0,0)
(B)(0.2, 0.4)
(C) (0.5,0.5)
(D) (1,2)
Two circles intersect at points satisfying both equations: (x – 0.5)² + y² = 0.25 and (x – 1)² + (y – 1)² = 1. The correct answer is (B) (0.2, 0.4), verified by substitution into both equations.
Circle Equations
The first circle has center (0.5, 0) and radius 0.5, so its equation is (x – 0.5)² + (y – 0)² = (0.5)² = 0.25. The second circle has center (1, 1) and radius 1, giving (x – 1)² + (y – 1)² = 1. Intersection points solve both simultaneously, and distance between centers is √((1-0.5)² + (1-0)²) = √1.25 ≈ 1.118, confirming two intersections since |1-0.5| < 1.118 < 1+0.5.
Option Verification
Verify each option by plugging into both equations (left side must equal right side).
| Option | First Circle Check | Second Circle Check | Result |
|---|---|---|---|
| (A) (0,0) | (0-0.5)² + 0² = 0.25 ✓ | (0-1)² + (0-1)² = 2 ≠ 1 ✗ | Incorrect |
| (B) (0.2,0.4) | (0.2-0.5)² + 0.4² = 0.09 + 0.16 = 0.25 ✓ | (0.2-1)² + (0.4-1)² = 0.64 + 0.36 = 1 ✓ | Correct |
| (C) (0.5,0.5) | (0.5-0.5)² + 0.5² = 0.25 ✓ | (0.5-1)² + (0.5-1)² = 0.25 + 0.25 = 0.5 ≠ 1 ✗ | Incorrect |
| (D) (1,2) | (1-0.5)² + 2² = 0.25 + 4 = 4.25 ≠ 0.25 ✗ | (1-1)² + (2-1)² = 1 ✓ | Incorrect |
Why (0.2, 0.4) Works
This point lies exactly on both circumferences, as distances match radii: from (0.5,0) is 0.5, from (1,1) is 1. The other intersection is approximately (1,0), but not listed. For full calculation, subtract equations to get the radical axis x + 2y = 1.5, then solve with one circle equation.
