Circles
Circle
nta_pyq_2025_jan
Grade 11
Question:
A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point (2, 5) and intersects the circle C at exactly two points. If the set of all possible values of r is the interval (\alpha, \beta), then 3\beta - 2\alpha is equal to :
Step-by-Step Solution
Key Concept: Apply the core result for circle equations and tangents and simplify using the given constraints.
(2) 2 2 C1 C2 = \sqrt(2 + 2) + (5 - 2) = \sqrt16 + 9 = 5 r + 2 > 5 r > 3 r < 5 + 2 r < 7 \therefore \alpha = 3, \beta = 7 3\beta - 2\alpha = 3(7) - 2(3) = 21 - 6 = 15
Correct Answer: 2