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Slopes of Tangent and Normal to the Curve

Here you will learn slopes of tangent and normal to the curve with examples.

Let’s begin –

Slopes of Tangent and Normal to the Curve

(a) Slopes of Tangent

Let y = f(x) be a continuous curve, and let P(x1,y1) be a point on it. Then, 

(dydx)P is the tangent to the curve y = f(x) at point P.

i.e. (dydx)P = tan ψ = Slope of the tangent at P,

where ψ is the angle which the tangent at P(x1,y1) makes with the positive direction of x-axis.

If the tangent at P is parallel to x-axis, then

ψ = 0 tan ψ = 0 Slope = 0 (dydx)P = 0

If the tangent at P is perpendicular to x-axis, or parallel to y-axis, then

ψ = π2 cot ψ = 0 1tanψ = 0 (dxdy)P = 0

(b) Slopes of Normal

The normal to the curve at P(x1,y1) is a line perpendicular to the tangent at P and passing through P.

  Slope of the normal at P = 1SlopeofthetangentatP = (dxdy)P

Example : find the slopes of the tangent and the normal to the curve x2+3y+y2 = 5 at (1, 1).

Solution : The equation of the curve is x2+3y+y2 = 5

Differentiating with respect to x, we get

2x + 3dydx + 2ydydx = 0

dydx = 2x2y+3

(dydx)(1,1) = -(22+3) = -25

  Slope of the tangent at (1, 1) = -25

and, Slope of normal at (1, 1) = 1slopeoftangentat(1,1) = 52


Related Questions

Show that the tangents to the curve y = 2x33 at the points where x =2 and x = -2 are parallel.

Find the slope of normal to the curve x = 1 – asinθ, y = bcos2θ at θ = π2.

Find the slope of the normal to the curve x = acos3θ, y = asin3θ at θ = π4.

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