This document defines tangent and normal lines to a curve and provides equations for tangent and normal lines. It also defines the length of the tangent, normal, sub-tangent, and sub-normal. Specifically: 1) A tangent line touches a curve at only one point, while a normal line is perpendicular to the tangent line at a given point. 2) The equation of a tangent line at a point is defined by the slope of the curve at that point. The equation of a normal line uses the inverse slope. 3) Formulas are provided for calculating the length of the tangent, normal, sub-tangent, and sub-normal in terms of the curve's equation.

1. Tangent And Normal
Presented By:- Ramesh Singh Makar Presented To:- Mr. Sunil Bhardwaj

2. 1. Definition
• Tangent
• Normal
2. Equation
• Tangent
• Normal
3. Length Of Tangent, Sub-Tangent, Length Of Normal, Sub-Normal
CONTENTS

3. Tangent :-A tangent is a line that touches a curve. A tangent meets or touches a
circle only at one point, whereas the tangent line can meet a curve at
more than one point.
(1) (2)
 On the other hand, a line may meet the curve once, but still not be a tangent

4. The normal line to a curve at a particular point is the line
through that point and perpendicular to the tangent.
Normal :-
X
Y
p(x₁ ,y₁)
.
T
Normal
Tangent
Y = f(X)
o

5. Equation Of Tangent
 Let Y = f(X) be the given Curve and p(x₁ , y₁) the given point on curve . Let PT be the
Tangent at the point p(x₁ , y₁).
Then the slope of the tangent to the curve is equal to dy/dx at point p.
Slope of Tangent PT = dy/dx (x₁ , y₁)
= dy₁/dx₁
Equation of Tangent to the Curve at p(x₁ ,y₁)
Y-y₁ = dy₁/dx₁ (X-x₁)
This is required equation at point p.
.
P(x₁ , y₁)
Y = f(X)
Tangent
Y
X
·
TO
𝑎2
+ 𝑏2
= 𝑐2

6. Equation Of Normal
 We Know that,
Normal is perpendicular to the Tangent at point p(x₁ , y₁) .
So…….
(slope of Tangent at p)(slope of Normal at p) = -1
m₁m₂ = -1 m₂ = -dx₁/dy₁
Required Equation of Tangent at point p(x₁ , y₁) Is……
Y-y₁ = (- dx₁/dy₁)(X-x₁)
Y = f(x)
Tangent
Normal
p(x₁ ,y₁)
o X
Y

7. Length of Tangent , Normal , Sub-
Tangent and Sub-Normal
X
Y
P(x,y)
Y =f(X)
GTo M
Let the Tangent and Normal at point p(x,y) to the Curve Y = f(X) meet the x-axis
at T and G respectively. And let PM with the ordinate through point P.

8. Here……
 PT is the Length of Tangent.
 PG is the Length of Normal.
 TM is the Length of Sub-Tangent .
MG is the Length of Sub-Normal.
1) Length of Tangent
In Triangle PMT
SinΨ = PM/PT = y/PT
PT = y/SinΨ = y CosecΨ
= y √(1+Cot²ψ)

9. Continue…….
PT = y√(1+1/tan²ψ)
= y√(1+tan²ψ)/tan²ψ
= y√{1+(dy/dx)²}/(dy/dx)
PT = y√1+(dx/dy)²
2) Length of Normal
In Triangle PMG
CosΨ = PM/PG = y/ PG

10. Continue…….
PG = y/Cosψ
PG = ySecψ
= y√(1+tan²ψ)
PG = y√1+(dx/dy)²
3) Length of Sub-Tangent
In Triangle PMT
tanψ = PM / TM
TM = y / (dy/dx)

11. 4) Length of Sub-Normal
In Triangle PMG
tan ψ = MG/ PM
Y₁ = MG/PG
MG = yy₁