If A and B are the points of intersection of the circle x + y - 8x = 0 and the hyperbola 2 y and a 2 2 x - = 1 9 4 point P moves on the line 2x - 3y + 4 = 0, then the centroid of △PAB lies on the line :
Step-by-Step Solution
Key Concept: Apply the core result for hyperbola parameters and tangents and simplify using the given constraints.
C: x + y - 8x = 0 H : 2 y 2 2 x - = 1 9 4 (3) By solving x 2 - ( 8x-x 2 ) = 1 9 4 2 2 4x - 72x + 9x = 36 2 \Rightarrow 13x - 72x - 36 = 0 2 \Rightarrow 13x - 78x + 6x - 36 = 0 \Rightarrow 13x(x - 6) + 6(x - 6) = 0 13 \Rightarrow x = 6 or - \times neglected 6 2 2 \Rightarrow y = 8(6) - (6) \Rightarrow y = $\pm$\sqrt12 So, points A and B are (6, \sqrt12), (6, -\sqrt12) 2h+4 P (h, ) 3 Centroid of △P AB is ( 12+h 2h+4 , ) 3 9 By options this centroid lies on the live 6x - 9y = 20
Correct Answer: 3