Equation of Tangents and Normal

At a given point on a curve, tangents and normal are almost like a straight line. At this point, a tangent is parallel to the curve, while the normal is perpendicular to it. 

Therefore, the tangent and normal equations can be evaluated similarly to any straight line. The lines associated with curves like a circle, parabola, ellipse, and hyperbola are known as tangents and normal. 

Because tangents and normal are straight lines, they are written as a linear equation in x and y coordinates. The general formula of the tangent and the normal equation is ax + by + c = 0. 

Before going through the equations of tangent and normal, let us first learn about tangent and normal briefly.

Properties of Tangent and Normal

The properties of tangents and normal are listed below:

• Normal and tangents are perpendicular to one another.
• The product of a tangent and a normal’s slope equals -1.
• Tangents are on the outside of the curve, while normal is on the inside.
• Every tangent of the curve has a normal associated with it.
• The normal curve may not surely travel through the focus or center of the curve.
• Straight lines, tangents, and normal are expressed as linear equations.
• A curve can have an endless number of tangents traced to it.

Formulas for Equation of Tangent and Normal questions

The formulas of tangent and normal to any curve at a point already given are as follows:

• |p  is the slope of the tangent to the curve y = f(x) at the point p.
• In a plane curve r = f(θ),

tanθ = r   

• The equation of the tangent at a point P (x1,y1) is

(y – y1) = p (x – x1)

• The equation of the normal at a point P (x1,y1) is

(x – x1) = p (y – y1)

Steps for the equation of the tangent

A tangent at a point on the curve is indeed a straight line that meets the curve and has the same slope as the curve’s derivative. You may derive how to obtain the equation of the tangent lines to the curve at any point from the equation.

Let us learn the steps with the help of an example:

Let the function y = f(x)

We have to find the tangent’s equation to the curve at x = x0. This can be found by following the steps given below:

• Step1: Using the formula-  m = dy/dx x = x0, we will find the derivative of the curve at the point x = x0.
• Step 2:  By using the formula m = y – y1/ x – x1, we find the equation of the straight line that is passing through the point (xo, y(x0) with slope m.

This will give you the equation of the tangent.

Steps for the equation of Normal

Let us find the equation of a normal line at point A (x1, y1)

Let the given curve’s equation = y = f(x)

• Step1: By using the equation that is already given, y = f(x), we will analyze (dy/dx)
• With the help of the formula given below, we will evaluate the slope to get the equation of the normal line to the curve at point A (x1, y1);

m= -1/ (dy/dx) x=x1; y=y1 

• In the final step, we replace the slope’s value in the (y – y1) = m(x – x1) equation.

Hence, we get the equation for the normal.

Equations of Tangent and Normal example

Consider the curve given by y = f(x) = x3 – x + 3.

1. Find the equation of the line tangent to the curve at the point (1, 3)
2. Find the line normal to the curve at the point (1, 3)
3. a) We can see that point (1,3) satisfies the curve’s equation. Now, for the equation of the tangent, we need the gradient of the curve at that point. It can be found as,

f(x) = x3 – x + 3

f′(x) = 3x2 – 1

Then, f′(x = 1) = 3. (1)2 – 1 = 2 = m

2. b) The tangent would be the straight line passing through (1,3) with slope = 2.

(y – 3) = 2(x – 1)

y = 2x + 1

2x – y + 1 = 0

Conclusion

The lines related to curves are tangents and normal. Each point on the curve has a tangent, a line that touches the curve at a particular location. A line perpendicular to the tangent at the point of contact is called normal. A straight line that meets the curve at a particular point is called a tangent to the curve. It does not intersect the curve but only touches the particular point already given. Normal, on the other hand, is a perpendicular straight line to the tangent.