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Question

Find the equation of a normal to the curve y = x loge x which is parallel to the line 2x − 2y + 3 = 0.

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Solution

Slope of the given line is 1
Let x1, y1 be the point where the tangent is drawn to the curve.Since, the point lies on the curve.Hence, y1= x1 loge x1 ... 1Now, y= x loge x dydx=x×1x+loge x 1=1+loge xSlope of tangent=1+loge x1Slope of normal = -1Slope of tangent=-11+loge x1Given thatSlope of normal = slope of the given line-11+loge x1=1-1=1+loge x1-2=loge x1x1=e-2=1e2Now, y1= e-2-2=-2e2 From (1)x1, y1=1e2, -2e2Equation of normal is,y+2e2=1 x-1e2y+2e2=x-1e2x-y=3e2x-y=3e-2

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