Circles
Circle
nta_pyq_2025_jan
Grade 11
Question:
Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C , whose mid-point is (1, 2), is :
\sqrt3
2\sqrt2
2\sqrt3
4\sqrt2
Step-by-Step Solution
Key Concept: Apply the core result for circle equations and tangents and simplify using the given constraints.
Let the centre be (3) (-2a, 6a-2 ) ≡ (-2a, 3a - 1) 2 Centre is equal distance from (4, 2) and (0, 2) 2 2 2 2 \Rightarrow \sqrt(4 + 2a) + (3a - 3) = \sqrt(-2a - 0) + (3a - 3) 2 2 2 2 \Rightarrow (2a + 4) + 9(a - 1) = 4a + 9(a - 1) 2 2 \Rightarrow 4a + 16 + 16a = 4a \Rightarrow a = -1 \Rightarrow centre ≡ (2, -4) \Rightarrow Radius = \sqrt40 2 2 2 \Rightarrow AM = (\sqrt40) - (\sqrt37) \Rightarrow 2AM = AB = 2\sqrt3
Correct Answer: 3