Hyperbola
Hyperbola
nta_pyq_2025_jan
Grade 11
Question:
Let the circle C touch the line x - y + 1 = 0, have the centre on the positive x -axis, and cut off a chord of length 2 2 y along the line -3x + 2y = 1. Let H be the hyperbola , whose one of the foci is the centre of C 4 x - = 1 2 2 \sqrt13 \alpha \beta and the length of the transverse axis is the diameter of C . Then 2\alpha + 3\beta is equal to ______ 2 2
Step-by-Step Solution
Key Concept: Apply the core result for hyperbola parameters and tangents and simplify using the given constraints.
(19) ∣ a+1 ∣ 2 2 r = \Rightarrow (a + 1) = 2r ∣ \sqrt2 ∣ 2 2 Also ( 3a-1 2 2 ) + ( ) = r \sqrt13 \sqrt13 2 2 3a - 1 4 (a + 1) \Rightarrow ( ) + = \sqrt13 13 2 2 5a - 14a - 3 = 0 1 \therefore a = - ,3 5 1 ∵ a \ne - \Rightarrow 5 \Rightarrow r = 2\sqrt2 2 2 y One focus of is (3, 0) x ∵ - = 1 2 2 \alpha \beta \Rightarrow \alphae = 3 and 2\alpha = 4\sqrt2 2 \Rightarrow \alpha = 2\sqrt2 \Rightarrow \alpha = 8 2 \beta 2 \alpha [1 + ] = 9 2 \alpha 2 2 \alpha + \beta = 9 2 \Rightarrow \beta = 1 2 2 \therefore 2\alpha + 3\beta = 19
Correct Answer: 19