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If the variable line 3x + 4y = a 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 a is _________.

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If the variable line 3x + 4y = a 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 a is _________.

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Both centres should lie on either side of the line as well as line can be tangent to circle.

(3 + 4 – α).(27 + 4 – α) < 0

(7 – α).(31 – α) < 0

⇒ α ∈ (7, 31) …(1)

d1 = distance of (1, 1) from line

d1 = distance of (9, 1) from line

d1 ≥ r1 ⇒ \(\frac{|7 - α|}{5}\) ≥ 1 ⇒ α ∈ (-∞, 2] ⋃ [12, ∞) ......(2)

d2 ≥ r2 ⇒ \(\frac{|31 - α|}{5}\) ≥ 2 ⇒ α ∈ (-∞, 21] ⋃ [41, ∞) ......(3)

(1) ⋂ (2) ⋂ (3) ⇒ α ∈ [12, 21]

Sum of integers = 165

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