Tangents and Normal

Slope of tangent and normal: The slope of the tangent to a continuous curve y = (x) at point p(x₁, y₁) is = 

 If the tangent of curve makes an angle θ with x-axis then

If tangent is parallel to x - axis then θ = 0

If it is perpendicular to x-axis (i.e. parallel to y-axis) then θ = π/2

 

Slope of normal at p = -  = - cot θ 

Equation of tangent: Tangent to the curve y = f(x) at point (x₁, y₁) pass through P and having slope 

∴ The equation of tangent at (x₁, y₁) to y = f(x) is

y - y₁ =  (x - x₁)

Equation of normal to y = f(x) at (x₁, y₁) is

y - y₁ =  (x - x₁)

y - y₁ = -  (x - x₁)

Note:

⇒ If  = ∞, Then tangent at P (x₁, y₁) is parallel to y-axis and its equation x = x₁

⇒ If  = 0 then the normal at p(x₁, y₁) is parallel to y-axis and its equation is x = x₁

⇒ The equation of tangent and normal to the curve having its parametric equation x = f(t)  and y = g(t) given by y - g(t) =  (x - f(t))… Equation of Tangent

Equation of normal is y - g(t) = -  (x - f(t))

Angle of intersection of two curves:

⇒ The angle of intersection of two curves defined to be the angle between the tangents to the two curves at their point of intersection.

⇒ Let C₁ and C₂ be two curves be 2 curves having equation y = f(x) and y = g(x) respectively. Let PPT₁ and PT₂ tangents to the curves C₁ and C₂ respectively at common point of intersection.

⇒ Then the angle between PT₁ and PT₂ is the angle of intersection of C₁ and C₂. Let θ₁ and θ₂ be  angles made PT₁ and PT₂ with positive direction of x-axis in anti-clock wise sense then m₁ = tan θ₁ = slope of tangent to y = f(x) at P.

⇒ = \[{{\left( \frac{dy}{dx} \right)}_{{{C}_{1}}}}\]

⇒  m₂ = tan θ₂ = slope of tangent to y = f(x) = 

∴ Angle between tangents is tan θ = 

The other angle between the tangents is 180 - θ

Generally the smaller of these two angles is taken to be the angle of intersection.