If the curves 2 x=y² and 2 x y=K intersect perpendicularly, then the value of K² is

Step-by-step solution

Given curves are 2 x=y²...(i) and 2 x y=K...(ii) On solving both Eqs. (i) and (ii), we get x= K^ 2 / 3 2 and y=K^ 1 / 3 Intersecting point of both curves is ( K^ 2 / 3 2 , K^ 1 / 3 ) Now differentiate Eq. (i) w.r.t. x , we get aligned 2 &=2 y d y d x d y d x &= 1 y aligned Slope of tangent at ( K^ 2 / 3 2 , K^ 1 / 3 )= 1 K^ 1 / 3 And differentiate Eq. (ii) w.r.t. x , we get 2 (x d y d x +y )=0 d y d x = -y x Slope of tangent at ( K^ 2 / 3 2 , K^ 1 / 3 )= -K^ 1 / 3 K^ 2 / 3 / 2 =-2 K^ -1 / 3 Since, both curves inter