If a variable line, is such that the two circles and are on its opposite sides, then the set of all values of is the interval
- A.[13, 23]
- B.(2, 17)
- C.[12, 21]
- D.(23, 31)
Show answer and solution
Correct option: C
Solution
The two circles have centers C₁ = (1,1) and C₂ = (9,1), with radii r₁ = 1 and r₂ = 2.
For the line 3x + 4y − λ = 0 to have the circles on opposite sides, the signed distances from each center must have opposite signs, AND the line must not intersect either circle (i.e., the distance from each center ≥ its radius).
Substituting C₁: 3(1) + 4(1) − λ = 7 − λ
Substituting C₂: 3(9) + 4(1) − λ = 31 − λ
For opposite sides: (7 − λ)(31 − λ) < 0, giving λ ∈ (7, 31).
Now apply the distance conditions:
- Distance from C₁ to line ≥ r₁: |7 − λ|/5 ≥ 1 → |7 − λ| ≥ 5 → λ ≤ 2 or λ ≥ 12
- Distance from C₂ to line ≥ r₂: |31 − λ|/5 ≥ 2 → |31 − λ| ≥ 10 → λ ≤ 21 or λ ≥ 41
Intersecting with λ ∈ (7, 31): from the first condition λ ≥ 12, from the second condition λ ≤ 21.
Therefore λ ∈ [12
For the line 3x + 4y − λ = 0 to have the circles on opposite sides, the signed distances from each center must have opposite signs, AND the line must not intersect either circle (i.e., the distance from each center ≥ its radius).
Substituting C₁: 3(1) + 4(1) − λ = 7 − λ
Substituting C₂: 3(9) + 4(1) − λ = 31 − λ
For opposite sides: (7 − λ)(31 − λ) < 0, giving λ ∈ (7, 31).
Now apply the distance conditions:
- Distance from C₁ to line ≥ r₁: |7 − λ|/5 ≥ 1 → |7 − λ| ≥ 5 → λ ≤ 2 or λ ≥ 12
- Distance from C₂ to line ≥ r₂: |31 − λ|/5 ≥ 2 → |31 − λ| ≥ 10 → λ ≤ 21 or λ ≥ 41
Intersecting with λ ∈ (7, 31): from the first condition λ ≥ 12, from the second condition λ ≤ 21.
Therefore λ ∈ [12
[JEE Main 2019 · 12 Jan · Shift I]