Tangent and Normal to a Curve

Let P be a point on the curve and Q (Q ≠ P) be the neighbouring point of P on the curve.

The line passing through P and Q is a secant of the curve.

The limiting position of the secant of the curve is known as a tangent of the curve.

Here, P is known as the point of contact.

Let us consider a curve, y = f(x).

Let P (a, b) be a point on the curve and let m be the slope of the tangent at this point.

The slope of the tangent to the curve, y = f(x), at a point,P(x_1,y₁), is m = (dy/dx)_p.

The slope of the tangent to the curve, y = f(x), at a point P(a,b) is (dy/dx)_p = f '(a).

The equation of a line passing through (x₁, y₁) with slope m is y - y₁ = m ( x - x₁).

The equation of the tangent to the curve, y = f(x), at P(a, b) is y - b = (dy/dx)_p (x - a).

⇒ (y - b) = f ' (a) (x - a)

If the tangent line to the curve, y = f(x), makes an angle 'θ' with the positive direction of the

X-axis, then the slope of the tangent, (dy/dx) = tan θ.

If (dy/dx)_p = 0, then the tangent line to the curve, y = f(x), is a horizontal line.

The equation of the tangent is y = b.

If (dy/dx)_p does not exist, then the tangent line at P to the curve, y = f(x), is a vertical line

The equation of the tangent is x = a.

A normal to a curve is a line that is perpendicular to the tangent at the point of contact.

Let 'm' be the normal to the curve at P.

Slope of the tangent, l = f¢(a)

⇒ Slope of normal m = -1/f ' (a)  [ product of the slopes of two perpendicular lines = -1]

The equation of the normal to the curve y = f(x) at the point P(a, b) is (y - b) = -1/f '(a) (x - a)

⇒ (y - b) f ' (a) + (x-a) = 0

(i) If (dy/dx)_p = f '(a) does not exist, then the normal at P (a, b) is parallel to the X-axis and its equation is y = b.

(ii) If (dy/dx)_p = f '(a) = 0, then the normal at (a, b) is parallel to the Y-axis and its equation is x=a.