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Maths-

General

Easy

Question

The equation of the normal line to the curve $y equals x log subscript e end subscript invisible function application x$ parallel to $2 x minus 2 y plus 3 equals 0$ is

$x + y = 3 e^{- 2}$
$x - y = 6 e^{- 2}$
$x - y = 3 e^{- 2}$
$x - y = 6 e^{2}$

The correct answer is: $x minus y equals 3 e to the power of negative 2 end exponent$

Given, $y equals x log subscript e end subscript invisible function application x$
$rightwards double arrow blank fraction numerator d y over denominator d x end fraction equals fraction numerator x over denominator x end fraction plus log invisible function application x equals 1 plus log invisible function application x$

$therefore$ Slope of normal $equals fraction numerator negative 1 over denominator fraction numerator d y over denominator d x end fraction end fraction equals fraction numerator negative 1 over denominator 1 plus log invisible function application x end fraction$

And given line is $2 x minus 2 y plus 3 equals 0 blank rightwards double arrow blank fraction numerator d y over denominator d x end fraction equals 1$

Since, normal line is parallel to the given line

$therefore blank minus fraction numerator 1 over denominator 1 plus log invisible function application x end fraction equals 1 blank rightwards double arrow blank x equals e to the power of negative 2 end exponent$

Now, intersecting point of given curve and $x equals e to the power of negative 2 end exponent$ is $left parenthesis e to the power of negative 2 end exponent comma blank minus 2 e to the power of negative 2 end exponent right parenthesis$

$therefore$ Required equation of line is

$y plus 2 e to the power of negative 2 end exponent equals 1 open parentheses x minus e to the power of negative 2 end exponent close parentheses blank rightwards double arrow blank x minus y equals 3 e to the power of negative 2 end exponent$

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