xy=0 then 2a xy 23. Try'+x® – 3axy=...
Question

# xy=0 then 2a xy 23. Try'+x® – 3axy=0 then dy is 22” xy (y? - ax} 223 (0) (a) none of these (a) (b) _ (y? – ax) (d) none of these

JEE/Engineering Exams
Maths
Solution
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( y^{3}+x^{3}-3 a x y=0 ) ( 3 y^{2} frac{d y}{d x}+3 x^{2}-3 a y-3 a x frac{d y}{d x}=0 ) ( left(y^{2}-a xright) frac{d y}{d x}+x^{2}-a y=0 ) ( y_{1}=-frac{left(x^{2}-a yright)}{y^{2}-a x} ) Differentiofe (1) again w- ( x ) th ( x ) ( left(2 y frac{d y}{d x}-aright) frac{d y}{d x}+left(y^{2}-a xright) frac{d y^{2}}{d x^{2}}+2 x-a frac{d y}{d x}=0 ) a) ( left(2 y y_{1}-aright) y_{1}+left(y^{2}-a xright) y_{2}+2 x-a y_{1}=0 ) a ( 2 y y_{1}^{2}-2 a y_{1}+2 x=-left(y^{2}-a xright) y_{2} ) ( Rightarrow y y_{1}^{2}-a y_{1}+x=-frac{left(y^{2}-a xright)}{2} y_{2} ) ( Rightarrow quad yleft(x^{2}-a yright)^{2}+aleft(x^{2}-yright)+x=-frac{left(y^{2}-a xright)}{y^{2}-a x}+y_{2} ) ( left(y^{2}-a xright)^{2} quad y^{2}-a x ) ( Rightarrow frac{y x^{4}+x y^{4}-3 a x^{2} y^{2}+a^{3} x y}{left(y^{2}-a xright)^{3}}=frac{-y_{2}}{2} ) ( begin{aligned} Rightarrow frac{-y_{2}}{2} &=frac{x yleft(x^{3}+y^{3}-3 a x yright)+a^{3} x y}{left(y^{2}-a xright)^{3}} &=frac{x y(0)+a^{3} x y}{left(y^{2}-a xright)^{3}}=frac{a^{3} x y}{left(y^{2}-a xright)^{3}} end{aligned} ) ( Rightarrow frac{d^{2} y}{d x^{2}}=frac{-2 a^{3} x y}{left(y^{2}-a xright)^{3}} ) option (a)