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Find the range of the parameter 'a' for which the line y = 2x + a lies strictly between the circles x² + y² - 2x - 2y + 1 = 0 and x² + y² - 16x - 2y + 61 = 0, without intersecting or touching either circle.

1. 2*sqrt(5) - 15 <= a <= -1 - sqrt(5)
2. -1 - sqrt(5) <= a <= 2*sqrt(5) - 15
3. sqrt(5) - 15 <= a <= 1 - sqrt(5)
4. 2*sqrt(5) - 15 <= a <= 1 + sqrt(5)

Correct answer: 2*sqrt(5) - 15 <= a <= -1 - sqrt(5)

Solution

Circle 1 is centred at (1,1) with r = 1 and circle 2 at (8,1) with r = 2; demanding the line lie between them and miss both gives 2*sqrt(5) - 15 <= a <= -1 - sqrt(5).

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