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Maths-

General

Easy

Question

Coordinates of a point on the curve $y equals x log invisible function application x$ at which the normal is parallel to the line $2 x minus 2 y equals 3$ , are

$(0, 0)$
$(e, e)$
$(e^{2}, 2 e^{2})$
$(e^{- 2}, - 2 e^{- 2})$

The correct answer is: $left parenthesis e to the power of negative 2 end exponent comma blank minus 2 e to the power of negative 2 end exponent right parenthesis$

Given curve is $y equals x$ log $blank x$
On differentiating w.r.t. $x$ , we get

$fraction numerator d y over denominator d x end fraction equals 1 plus log invisible function application x$

The slope of the normal $equals negative fraction numerator 1 over denominator left parenthesis d y divided by d x right parenthesis end fraction equals fraction numerator negative 1 over denominator 1 plus log invisible function application x end fraction$

The slope of the given line $2 x minus 2 y equals 3$ is1

Since, these lines are parallel

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

and $y equals negative 2 e to the power of negative 2 end exponent$

$therefore$ Coordinates of the point are $left parenthesis e to the power of negative 2 end exponent minus 2 blank comma blank minus 2 e to the power of negative 2 end exponent right parenthesis$

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