Straight line

“Geometry as a logical system is a method, even the most powerful means, to make youngsters realise the strength of the human spirit, which is their own spirit,” H. Freudenthal said. We all are very much familiar with geometry, to be precise two dimensional one – mainly it is a combination of algebra and geometry. Rene Descartes was the first mathematician to study this aspect. We all are familiar with plotting of points in a plane, distance between two points, section formulae and various other basic concepts of coordinate geometry. Here we shall look at topics like straight line, locus, equation of locus etc. 

Definition and basics of straight line.

The simplest locus in a plane is called a line. Now in a locus if we find the slope of a segment joining any two points on this locus then the slope is constant. A straight line always measures 180˚ angle. If a line meets the X axis in the point A (a,0) then ‘a’ is called X-intercept of the line. If it meets the Y-axis in the point B(0,b) then ‘b’ is known as the y intercept of that line. 

Locus

Any set of points in a plane that satisfies certain geometrical conditions or just a condition is known as locus. Denoted as follows:

L = { P│P is a point in the plane and P satisfies given geometrical condition}

Here p = representative of all points in L. 

L is called the locus of point P. Locus is a set of points.

The locus in an alternate way can also be described as the route of a point which moves while satisfying required conditions.

The equation is said to be the locus equation if the set of points whose coordinates fulfil a given equation in x and y is the same as the set of points on a locus.

Example:  Let L = {P │OP =4 } . Find the equation of L.

Solution: L is locus of points in the plane which are at 4 units distance from the origin

Let P(x , y) be any point on the locus L.

As op = 4 , op2 = 16

(x – 0)2 + (y -0)2 = 16

X2 + Y2  = 16

This above equation is the locus equation.

The locus is seen to be a circle.

Slope of a line

The smallest angle made by a line with the positive direction of the X-axis measured in counter clockwise sense is known as inclination of the line. We denote inclination of the line by θ.

Now if the inclination of a line is θ then tan θ is known as the slope of the line. It is denoted as m 

Mathematically m is expressed as:

m = tan θ

and tan θ = y2–y1/x2–x1 , where x and y are the coordinates.

One can say that two lines are parallel if and only if they have the same slope.

Example: find the slope of the line which passes through the points A(2,4)  and B(5,7).

Solution: the slope of the line passing through the points A(x1 , y1) and B(x2 , y2) is given by 

m = y2–y1/x2–x1 = 7-45-2 =33 = 1

Perpendicular lines

The coordinate axes are perpendicular to one another, just as a horizontal and vertical line are perpendicular to one another. One of them has a zero slope, while the other has an undefined slope. Let’s see if there’s a relationship between the slope of non-vertical lines and the slope of vertical lines.

Theorem: Non-vertical lines have slopes m1 and m2 and are perpendicular to each other if and only if m1*m2 = -1

Example:

1. Show that line AB is perpendicular to the line BC where A(1,2) , B(2,4) and C(0,5).

Solution: let slopes of line AB and BC be m1 and m2 respectively.

Therefore, m1 = 4-22-1 = 2 

And m2 = 5-40-2 = –12

Now, m1*m2 = 2 * -12 = -1

Since it satisfies our theorem, we can say that line AB is perpendicular to line BC.

Angle between intersecting lines

Till now we have worked out on the relation between slopes of perpendicular lines but what If the two given lines are not perpendicular? the question arises how to find the angle between them?

Well, a theorem already exists for such cases. let’s have a look at the below theorem

Theorem: if θ is the acute angle between non-vertical lines have slopes m1 and m2 then 

Tan θ = │m1–m21+m1m2│

Example: find the acute angle between lines having slope 3 and -2

Solution:  let m1 = 3 and m2 = -2

We know that, if θ is the acute angle between non-vertical lines have slopes m1 and m2 then 

Tan θ = │m1–m21+m1m2│

Tan θ = │3+21-6│ = 1 

Therefore, θ = 45˚

So, the acute angles between line having slope 3 and -2 is 45˚

Equation of line in standard form

If there is an equation in x and y which is satisfied by the coordinates of all points of a line and no other points then it is called the equation of the line.

• Every point on the x-axis has a Y coordinate of 0, and this is true exclusively for points on the X axis. As a result, the X axis equation is y =0.
• Similarly, the equation of the y axis is x=0
• The equation of any line parallel to the Y-axis is of the type x =k , (k is constant) and the equation of any line parallel to the X-axis is of the type y =k.

The equation of line in point slope form is given as: y- y1=m( x- x1)

Example: find the equation of the line passing through the point A(2,1) and having slope -3

Solution: we know that,

 The equation of line in point slope form is given as: y- y1=m( x- x1)

y – 1 = -3( x -2)

y-1 =-3x +6

3x +y -7 = 0 

This is the equation of line.

Conclusion

The simplest locus in a plane is called a line. A straight line always measures 180˚ angle. If the set of points, whose coordinates satisfy a certain equation in x and y, is the same as the set of points on a locus, then the equation is said to be the locus equation. If the inclination of a line is θ then tan θ is known as the slope of the line and

tan θ = y2–y1/x2–x1 ,

where x and y are the coordinates. The equation of line in point slope form is given as:

y- y1=m( x- x1).