Notes On Tangent and Normal to a Curve - CBSE Class 12 Maths
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(x1,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.

#### Summary

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(x1,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.

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