With usual notations in ABC , if B = 2 , and A 2 , C 2 are roots of equation p x^2+ q x+ r =0 , p 0 , then

Step-by-step solution

In the right triangle ABC with B = 2 , the sum of the remaining angles gives A + C = 2 , so A 2 + C 2 = 4 . Taking tangent on both sides and applying the tangent addition formula yields A 2 + C 2 1 - A 2 C 2 = 1 , which simplifies to A 2 + C 2 + A 2 C 2 = 1 . Given that A 2 and C 2 are roots of px^2 + qx + r = 0 , by Vieta's formulas, their sum is - q p and product is r p , so - q p + r p = 1 . Multiplying through by p and rearranging gives the relation p + q = r . Answer: A