question_answer 1)

Prove that the curve x = y2 and xy = K cut at right angles if 8K2 = 1. 

Answer:

Given curve are             x = y2                ?. (1)             xy = K              ?.. (2)                       [by (1)]        y = K1/3       (1)  x = K2/3              Point of intersection of (1) and (2) is P(K2/3, K1/3)          Diff. (1) w.r.t. x                         m1 = slope of tangent to (1)             Diff.(2) w.r.t. x             m2 = slope of tangent to (2)                   Acc. to Question       m1m2 = ? 1                                                             Cubing both sides                         8K2 = 1             Which is required result.