If x^3+y^3=3 a x y , then at ( 3 a 2 , 3 a 2 ) the value of 3 a y^ +40 is

Step-by-step solution

x^3+y^3=3 a x y3 x^2+3 y^2 d y d x =3 a y+3 a x d y d x aligned & (y^2-a x ) d y d x =a y-x^2 & d y d x = a y-x^2 y^2-a x ( d y d x ) ( 3 a 2 , 3 a 2 )=-1 & (2 y d y d x -a ) d y d x + (y^2-a x ) d^2 y d x^2 =a d y d x -2 x & 2 y ( d y d x )^2+ (y^2-a x ) d^2 y d x^2 =2 a d y d x -2 x aligned At ( 3 a 2 , 3 a 2 ) ; 2 3 a 2 (-1)^2+ ( 3 a^2 4 ) ( d^2 y d x^2 )=2 a(-1)-3 a aligned & 3 a+ 3 a^2 4 ( d^2 y d x^2 )=-5 a & 3 a 4 ( d^2 y d x^2 )=-8 & 3 a y^ =-32 & 3 a y^ +40=40-32=8 aligned