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Tangents And Normals

Tangents and normals are the lines associated with curves. The tangent is a line touching the curve at a distinct point, and each of the points on the curve has a tangent. Normal is a line perpendicular to the tangent at the point of contact. The equation of the talent at the point (x₁, y₁) is of the form (y - y₁) = m(x - x₁), and the equation of a normal passing through this same point is (y - y₁) = -1/m. (x - x₁).

Let us learn more about how to find the equation of tangents and normals for different curves such as a circle, parabola, ellipse, hyperbola, and their properties with the help of examples, FAQs.

1.	What Are Tangents And Normals?
2.	How To Find Tangents And Normals?
3.	Tangents And Normals For Different Curves
4.	Properties Of Tangents And Normals
5.	Examples On Tangents And Normals
6.	Practice Questions
7.	FAQs On Tangents And Normals

What Are Tangents And Normals?

Tangents and normals are the lines associated with curves such as a circle, parabola, ellipse, hyperbola. A tangent is a line touching the curve at one distinct point, and this distinct point is called the point of contact. Normal is a line perpendicular to the tangent, at the point of contact. The normal is also passing through the focus of the curve.

There are numerous tangents that can be drawn to a curve, at each of the distinct points lying on the curve. The tangents and normals are straight lines and hence they are represented as a linear equation in x and y. The general form of the equation of a tangent and normal is ax + by + c = 0. The point of contact satisfies the equation of the tangent and the equation of the curve.

How To Find Tangents And Normals?

The tangent and normal can be computed with the help of the equation of the curve. The equation of a tangent and normal can be computed by the differentiation of the equation of the curve. The differentiation of the curve with respect to the independent variable x is dy/dx and it gives the slope of the tangent, and the negative inverse of the differentiation -dx/dy gives the slope of the normal to the curve.

This slope is represented as m = dy/dx, and the equation of the tangent and normal can be calculated with the help of the point-slope form of the equation of the line - (y - y₁) = m(x - x₁).

The tangent and the normal are perpendicular to each other, and the product of the slope of the tangent and the slope of the normal is equal to -1. The general form of the equation of a tangent passing through a point (x₁, y₁) and having a slope m is \((y - y_1) = m(x - x_1)\). And the equation of a normal passing through this same point is \((y - y_1) = \dfrac{-1}{m}(x - x_1)\)

Tangents And Normals For Different Curves

The tangents and normals can be formed for the following below set of curves. The tangents and normals can be drawn for a circle, parabola, ellipse hyperbola, respectively.

Circle: The equation of a tangent to a circle x² + y² + 2gx + 2fy + c = 0 at the point (x₁, y₁) is xx₁ + yy₁ + g(x + x₁) + f(y + y₁) + c = 0.
Parabola: The equation of tangent to a parabola y² = 4ax, at the point (x₁, y₁), is yy₁ = 2a(x + x₁).
Ellipse: The equation of tangent to a ellipse x²/a² + y²/b² = 1 at the point (x₁, y₁) is xx₁/a² + yy₁/b² = 1.
Hyperbola: The equation of tangent to a hyperbola x²/a² - y²/b² = 1 at the point (x₁, y₁) is xx₁/a² - yy₁/b² = 1.

The equation of the normal at the point (x₁, y₁), for each of the curve, can be calculated by taking the inverse of negative differentiation of the curve at the point, as the slope and then forming the equation of the normal.

Properties Of Tangents And Normals

The following properties of tangents and normals help in a better understanding of tangents and normals.

Tangent and normals are perpendicular to each other.
The product of the slopes of a tangent and a normal is equal to -1.
Tangents lie outside the curve and normals lie inside the curve.
A normal is associated with every tangent of the curve.
The normal to curve may not surely pass through the focus or center of the curve.
Tangents and normals are straight lines and are represented as linear equations.
There is an infinite number of tangents that can be drawn to a curve.

Related Topics

The following topics help for a better understanding of tangents and normals.

Examples on Tangents And Normals

Example 1: Find the equation of tangent and normal to the circle x² + y² = 5, at the point (2, 3).

Solution:

The given equation of the circle is x² + y² = 5.

The slope of the tangent is obtained by taking the derivative of the above expression with respect to x.

2x + 2y.dy/dx = 0

dy/dx = -2x/2y

dy/dx = -x/y.

Let us substitute the point (2, 3) in the above differentiation to obtain the slope of the tangent.

Slope of tangent = m = dy/dx = -2/3

The equation of the tangent can be computed using the point slope form of equation of line - (y - y₁) = m(x - x₁).

(y - 3) = -2/3.(x - 2)

3(y - 3) = -2(x - 2)

3y - 9 = -2x + 4

2x + 3y -9 - 4 = 0

2x + 3y -13 = 0

2x + 3y = 13.

The slope of the normal is m = -dx/dy = -(-y/x) = y/x = 3/2

Equation of the normal can also be computed using the point(2,3), and through the point slope form of equation of lne - (y - y₁) = m(x - x₁).

(y - 3) = 3/2(x - 2)

2(y - 3) = 3(x - 2)

2y - 6 = 3x - 6

3x - 2y = 0

Therefore the equation of the tangent is 2x + 3y = 13, and the equation of the normal is 3x - 2y = 0.
Example 2: Find the equation of tangent and normal to the curve, \(y = \dfrac{(x - 5)}{(x - 2)(x - 3)}\),where the curve cuts the x-axis.

Solution:

The given equation of the curve is \(y = \dfrac{(x - 5)}{(x - 2)(x - 3)}\).

The point where the above curve cuts the x-axis can be obtained by substituting y = 0 in the above equation.

\(0= \dfrac{(x - 5)}{(x - 2)(x - 3)}\)

x - 5 = 0

x = 5

Hence the curve cuts the x-axis at the point (5, 0).

The equation of the tangent and normal can be calculated by first calculating the slope of the above curve at the point (5, 0)

\(\dfrac{dy}{dx} = \dfrac{(x^2 - 5x + 6).dy/dx(x - 5) - (x -5).dy/dx.(x^2 - 5x + 6)}{(x^2 - 5x + 6)^2}\)

\(\dfrac{dy}{dx} = \dfrac{(x^2 - 5x + 6).1 - (x - 5)(2x - 5)}{((x^2 - 5x + 6)^2}\)

Let us substitute the point (5, 0) in the above expression to obtain the slope of the tangent.

\(m = \dfrac{dy}{dx} = \dfrac{(5^2 - 5.5 + 6).1 - (5 - 5)(2.5 - 5.5)}{(5^2 - 5.5 + 6)^2}\)

\(m = \dfrac{dy}{dx} = \dfrac{6}{36}\)

m = dy/dx = 1/6

Let us find the equation of the tangent having the slope m = 1/6, and passing through the point (5, 0).

(y - 0) = 1/6 (x - 5)

6y = x - 5

x - 6y - 5 = 0

The equation: x - 6y = 5, is the required equation of the tangent to the curve.

The slope of the normal is m = -dx/dy = -(6/1) = -6

The equation of the normal having the slope m = -6 and passing through the point (5, 0) is as follows.

(y - 0) = -6(x - 5)

y = -6(x - 5)

y = -6x + 30

The equation, 6x + y = 30 is the required equation of the normal.

Therefore the equation of the tangent to the curve is x - 6y = 5, and the equation of the normal is 6x + y = 30.

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Practice Questions on Tangents And Normals

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FAQs on Tangents And Normals

What Are Tangents And Normals?

Tangents and normals are the lines associated with curves such as a circle, parabola, ellipse, hyperbola. A tangent is a line touching the curve at one distinct point, and this distinct point is called the point of contact. Normal is a line perpendicular to the tangent, at the point of contact. The normal is also passing through the focus of the curve.

How To Find Tangents And Normals?

The tangents and normals can be found with the help of the point-slope form of the equation of the line - (y - y₁) = m(x - x₁). The slope of the tangent can be computed by taking the derivative of the equation of the curve with respect to the independent variable x, which is equal to m = dy/dx, and the slope of the normal is equal to the negative inverse of the differentiation m = -dx/dy.

What Is The Equation Of Tangents And Normals?

The equation of tangent and normal can be computed through the coordinate geometry formal of point-slope form. The equation of the tangent is of the form (y - y₁) = m(x - x₁), and the equation of a normal passing through this same point and perpendicular to the tangent is (y - y₁) = -1/m. (x - x₁).

What Do Tangents And Normals Tell Us?

The tangents tells us the direction in which the force acting on a body moving in a circular motion would act outwards, and the same holding force acting on the body in the inward direction is identified with the normal.

What Is The Relationship Between Tangents And Normals?

The tangents and normas are the set of lines that are perpendicular to each other. The product of the slopes of the tangents and normals is equal to -1. The slope of the tangent is m = dy/dx, and the slope of the normal is m = -dx/dy.

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