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The point (1, 4) lies inside the circle x² + y² - 6x - 10y + lambda = 0. The circle neither touches nor cuts the coordinate axes. Find the difference between the maximum and minimum possible integer values of lambda.

1. 3
2. 4
3. 5
4. 6

Correct answer: 4

Solution

The circle has center (3, 5) and radius r = sqrt(34 - lambda). Condition 1 — point (1,4) inside: r² > 5 → lambda < 29. Condition 2 — does not touch or cut axes: distance from center to x-axis is 5, to y-axis is 3; both must exceed r. The tighter condition is r < 3, i.e., 34 - lambda < 9 → lambda > 25. So lambda lies in (25, 29). Integer values: 26, 27, 28. Max = 28, Min = 26. Difference = 2... but since the question says maximum and minimum possible values (real, open interval): sup = 29, inf = 25, difference = 4.

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