If y²=a x²+b x+c, where a, b, c are constants, then y³ d² y d x² is equal to

Step-by-step solution

Given, y²=a x²+b x+c On differentiating w.r.t. x , we get 2 y d y d x =2 a x+b Again differentiating w.r.t. x , we get aligned & 2 ( d y d x )²+2 y d² y d x² =2 a & y d² y d x² =a- ( d y d x )² aligned y d² y d x² =a- ( 2 a x+b 2 y )² y d² y d x² = 4 a y²-(2 a x+b)² 4 y² 4 y³ d² y d x² =4 a (a x²+b x+c )- (4 a² x²+4 a b x+b² ) 4 y³ d² y d x² =4 a c-b² y³ d² y d x² = 4 a c-b² 4 = constant