JEE Main202430 Jan 2024Morning ShiftMathematicsCircleActual
If the circles ( x + 1 ) 2 + ( y + 2 ) 2 = r 2 and x 2 + y 2 - 4 x - 4 y + 4 = 0 intersect at exactly two distinct points, then
Options
- A5 < r < 9
- B0 < r < 7
- C3 < r < 7
- D1 2 < r < 7
Correct answer
C. 3 < r < 7
Step-by-step solution
Given, S 1 ≡ x + 1 2 + y + 2 2 = r 2 and S 2 ≡ x 2 + y 2 - 4 x - 4 y + 4 = 0 or S 2 ≡ x - 2 2 + y - 2 2 = 2 2 Now, if two circles intersect at two distinct points then r 1 - r 2 < C 1 C 2 < r 1 + r 2 . ⇒ r - 2 < 3 2 + 4 2 < r + 2 ⇒ r - 2 < 5 < r + 2 ⇒ r - 2 < 5 and r + 2 > 5 ⇒ - 5 < r - 2 < 5 and r > 3 ⇒ - 3 < r < 7 and r > 3 ⇒ 3 < r < 7