Normal to a Curve

The normal to the curve is the line perpendicular (at right angles) to the tangent to the curve at that point. Its equation can be found out by the point-slope form as the tangent's point of contact and slope will be known.

If a line with slope 1 is tangent to a curve at the point \((1,1),\) find the equation of the normal at the curve at that point.

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The slope of the normal is \(-1.\) \((\)For perpendicular lines, the product of slopes is \(-1.)\) We know that \((1,1)\) lies on it. Thus the equation is \(y=mx+c\), where \(m\) is \(-1.\) Satisfy \((1,1)\) in the equation: \(1=-1+c \implies c=2.\) Thus the equation of the normal to the curve at that point is \(y=-x+2,\) or \(x+y=2.\ _\square\)