Tangent and normal

Mathematics

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

2. Definition Tangent Normal Equation Tangent Normal 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) = dy ₁/ dx ₁ Equation of Tangent to the Curve at p(x₁ ,y₁) Y-y₁ = dy ₁/ dx ₁ (X-x₁) This is required equation at point p. . P(x₁ , y₁) Y = f(X) Tangent Y X · T O  

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 m₁m ₂ = -1 m₂ = - dx ₁/ dy ₁ Required Equation of Tangent at point p(x₁ , y₁) Is…… Y-y₁ = (- dx ₁/ dy ₁)(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) G T ₼ o 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 Sin Ψ = PM/PT = y/PT PT = y/Sin Ψ = y Cosec Ψ = y √(1+Cot ² ψ )

9. Continue……. PT = y√(1+1/tan² ψ ) = y√(1+tan² ψ )/tan² ψ = y√{1+( dy / dx )²}/( dy / dx ) PT = y√1+( dx / dy )² Length of Normal In Triangle PMG Cos Ψ = PM/PG = y/ PG

10. Continue……. PG = y/Cos ψ PG = ySec ψ = y √( 1+tan² ψ ) PG = y√1+(dx/ dy )² 3) Length of Sub-Tangent In Triangle PMT tan ψ = PM / TM TM = y / ( dy /dx)

11. Length of Sub-Normal In Triangle PMG tan ψ = MG/ PM Y₁ = MG/PG MG = yy ₁