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08/02/2022 2:09 pm
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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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08/02/2022 2:42 pm
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