$If xy - {log}_{e} y = 1 satisfies the equation and x ({yy}_{2} + y_{1}^{2}) - y_{2} + k {yy}_{1} = 0, then the value of k is$

-3

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-1

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Solution

The correct option is C 3
$We have, xy - {log}_{e} y = 1 Differentiating w . r . t . x {xy}_{1} + y - \frac{1}{y} y_{1} = 0 {xyy}_{1} + y^{2} - y_{1} = 0 Again differentiating w . r . t . x {xyy}_{2} + {xy}_{1}_{1}^{2} + {yy}_{1} + 2 {yy}_{1} - y_{2} = 0 x ({yy}_{2} + {y_{1}}_{1}^{2}) - y_{2} + 3 {yy}_{1} = 0 \Rightarrow k = 3$

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