For a>0, t (0, 2 ) , let x= a^ ⁻¹ t and y= a^ ⁻¹ t , Then, 1+ ( d y d x )^2 equals :

Step-by-step solution

aligned & Let x= a^ ⁻¹ t & x^2=a^ ⁻¹ t 2 x= ⁻¹ t a & 2 x = a 1-t^2 d t d x & 2 1-t^2 x a = d t d x aligned Now, let y= a^ ⁻¹ t aligned & 2 y= ⁻¹ t a & 2 y d y d x = - a 1-t^2 d t d x aligned 2 y d y d x = - a 1-t^2 2 1-t^2 x a (from(1) d y d x =- y x Hence, 1+ ( d y d x )^2=1+ ( -y x )^2= x^2+y^2 x^2