If the variable line 3 x + 4 y = α lies between the two circles ( x - 1 ) 2 + ( y - 1 ) 2 = 1 and ( x - 9 ) 2 + ( y - 1 ) 2 = 4 , without intercepting a chord on either circle, then the sum of all the integral values of α is

Question

If the variable line 3 x + 4 y = α lies between the two circles ( x - 1 ) 2 + ( y - 1 ) 2 = 1 and ( x - 9 ) 2 + ( y - 1 ) 2 = 4 , without intercepting a chord on either circle, then the sum of all the integral values of α is,

MARKS App · Accepted Answer

Given line 3 x + 4 y = α Given circles ( x - 1 ) 2 + ( y - 1 ) 2 = 1 & ( x - 9 ) 2 + ( y - 1 ) 2 = 4 According to the given information centres of both the circles must lie on opposite side of line. L 11 . L 22 0 ⇒ 3 + 4 - α 27 + 4 - α 0 α ∈ 7 , 31 . Line is neither touching nor intersecting that means perpendicular distance from centre to line must more than or equal to radius. ⇒ 3 + 4 - α 5 ≥ 1 & 27 + 4 - α 5 ≥ 2 ⇒ α ∈ ( - ∞ , 2 ] ∪ [ 12 , ∞ ) & α ∈ ( - ∞ , 21 ] ∪ [ 41 , ∞ ) ⇒ α ∈ 12 , 21 Sum of integer values of α = 12 + 13 + - - 21 = 165 .

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