Question

If xy − log_e y = 1 satisfies the equation $x\left(y{y}_2+y_1^2\right)-y_2+\lambda y{y}_1=0$, then λ =

(a) −3
(b) 1
(c) 3
(d) none of these

Answer 1

(c) 3

Here,

$$\begin{gathered} xy-{\log }_{e}y=1 \\ \Rightarrow x{y}_1+y-\frac{y_1}{y}=0 \\ \Rightarrow xy{y}_1+y^2-y_1=0 \\ \Rightarrow y{y}_1+x{y}_1{y}_1+xy{y}_2+2y{y}_1-y_2=0 \\ \Rightarrow x\left({y_1}^2+y{y}_2\right)-y_2+3y{y}_1=0 \\ \therefore \lambda =3 \end{gathered}$$