Analyzing Normal and Tangent examples

Tangents and normal can be defined as lines that are associated with different types of curves. The tangent refers to a line touching the curve at a particular point. This also means that each point on the curve has a particular tangent. On the contrary, the line of normal is perpendicular to the tangent at that particular point. The tangent’s equation at a particular point (x1, y1) on the curve is given as (y – y1) = m(x – x1). On the other hand, the equation of normal at that point is given as (y – y1) = -1/m(x – x1).

Tangents and normal definition

Tangents and normal can be described as some specific lines associated with points on certain curves such as hyperbola, ellipse, circle, and hyperbola. Specifically, the tangent refers to a line touching a particular curve at one specific point. On the contrary normal is a line that is perpendicular to the given tangent at that specific point of contact. An important property of normal is that it passes through the curve’s focus. Separate tangents can be drawn at each distinct point on a given curve. The normals and tangents are straight lines. For this reason, they are shown as linear equations in y as well as x. The general form of the equation of tangent and normal is bx + cy + d = 0. In this context, it should be mentioned that the tangent’s equation and the curve’s equation are satisfied by the point of contact. 

Tangents and normal formula

In mathematics, we can see that there are several formulas through which the tangent and normals to a particular curve can be calculated. These tangent and normal formulas have been outlined in the following. 

1. Suppose y = f(x) is a curve and the slope of the tangent to this curve at a particular point p on the curve is given as dy/dx|p 
2. If we consider a plane curve in the form r = f(θ) then we obtain tan ϕ = r * dθ / dr.
3. The tangent’s equation at the point P (x2, y2) is given as 

(y – y2) = dy/dx|p (x – x2).

1. The normal’s equation at the point of contact P (x2, y2) is given as 

(x – x2) = dy/dx|p (y – y2).

1. Let us suppose that a particular curve c is given as y = f(x). Moreover suppose that ‘s’ is the length of the perpendicular from the origin (0,0) to the point to the curve’s tangent at the point (x2,y2) of the curve. Then s = |y2 – x2dy/dx|/√[1 – (dy/dx)2].

Equations of tangent and normal

Generally, derivatives are utilized for finding the slope of a curve. Let the equation of a curve be y = f(x). Then the slope at the point b on the curve is given as f’(b) (the derivative of the function f(x) at the point b). Hence the equation of tangent and normal in the point-slope form is given as (y – f(b))/ (x – b) = f’(b) and (y – f(b))/ (x – b) = – 1/ f’(b) respectively. Here the slope of the normal is obtained from the equation mt * mn = -1.

Examples of Tangent and Normal

Example 1: Determine the equation of normal and tangent to the circle y2 + x2 = 4 at point (1, 1). 

Solution: The given circle’s equation is y2 + x2 = 4.

Here the slope of the tangent is dy/ dx = -x/y. 

If substitute we the point (1, 1) in the above slope, then we obtain dy/dx = -1.

Therefore the equation of tangent is given as 

(y – 1) = -1 (x – 1)

y – 1 = 1 – x

x + y = 2

Therefore the equation of normal is 

(y – 1) = -1/ (-1) (x – 1)

y – 1 = x – 1

x – y = 0 

Hence the equation of tangent and normal are x + y = 2 and x – y = 0 respectively. 

Conclusion

The entire article has been written on the core topic of examples of tangent and normal. Tangent and normal an important concepts in mathematics. The article has been thoroughly discussed throughout the article discussions on the definition of tangent and normal, Tangent and normal formula, equations of tangent and normal, and examples of tangent and normal.