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 = 2x3–3 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.