Two circles and are given. Then the equation of the circle through their points of intersection and the point is
- A.
- B.
- C.
- D.None of these
Show answer and solution
Correct option: B
Solution
The family of circles through the intersection of the two given circles is:
(x² + y² - 6) + λ(x² + y² - 6x + 8) = 0
Dividing and collecting: (1+λ)(x² + y²) - 6λx + (8λ - 6) = 0, or x² + y² - (6λ/1+λ)x + (8λ-6)/(1+λ) = 0.
Since (1,1) lies on it: substitute x=1, y=1:
(1+λ)(1+1) - 6λ(1) + (8λ-6) = 0
2 + 2λ - 6λ + 8λ - 6 = 0
4λ - 4 = 0 → λ = 1.
Substituting λ=1: 2(x²+y²) - 6x + 2 = 0, i.e., x² + y² - 3x + 1 = 0, which is option 2.
(x² + y² - 6) + λ(x² + y² - 6x + 8) = 0
Dividing and collecting: (1+λ)(x² + y²) - 6λx + (8λ - 6) = 0, or x² + y² - (6λ/1+λ)x + (8λ-6)/(1+λ) = 0.
Since (1,1) lies on it: substitute x=1, y=1:
(1+λ)(1+1) - 6λ(1) + (8λ-6) = 0
2 + 2λ - 6λ + 8λ - 6 = 0
4λ - 4 = 0 → λ = 1.
Substituting λ=1: 2(x²+y²) - 6x + 2 = 0, i.e., x² + y² - 3x + 1 = 0, which is option 2.
[1980]