DURGESH KUMAR SINGH
ROLL NO. R355A06
History of all great works is to witness that no great work
was ever done without either the active or passive support
a person‘s surrounding and one’s close quarters. Thus it is
not hard to conclude how active assistance from seniors.
Could prohibitively impact the execution of a project .I am
highly thankful to our learned faculty Mr. Pankaj Kumar
for her active guidance throughout the completion of
Last but not the least, I would also want extend my
appreciation to those who could not be mentioned here but
here well played their role to inspire the curtain.
Durgesh Kumar Singh
3. Coordinates of mid point
4. Distance between two points
5. Section formula
6. Coordinates of centroid
7. Area of triangle
8. Straight lines
9. Slope of straight lines
10. General equation of straight line
11. Angle between two straight lines
12. Length of perpendiculars from a point
13. Uses of coordinate geometry
The Greek mathematician Menaechmus, solved problems and proved theorems by using a
method that had a strong resemblance to the use of coordinates and it has sometimes been
maintained that he had analytic geometry. Apollonius of Perga, in On Determinate Section
dealt with problems in a manner that may be called an analytic geometry of one dimension; with
the question of finding points on a line that were in a ratio to the others. Apollonius in the
Conics further developed a method that is so similar to analytic geometry that his work is
sometimes thought to have anticipated the work of Descartes by some 1800 years. His
application of reference lines, a diameter and a tangent is essentially no different than our
modern use of a coordinate frame, where the distances measured along the diameter from the
point of tangency are the abscissas, and the segments parallel to the tangent and intercepted
between the axis and the curve are the ordinates. He further developed relations between the
abscissas and the corresponding ordinates that are equivalent to rhetorical equations of curves.
However, although Apollonius came close to developing analytic geometry, he did not manage
to do so since he did not take into account negative magnitudes and in every case the coordinate
system was superimposed upon a given curve a posteriori instead of a priori. That is, equations
were determined by curves, but curves were not determined by equations. Coordinates, variables,
and equations were subsidiary notions applied to a specific geometric situation.
The eleventh century Persian mathematician Omar Khayyám saw a strong relationship between
geometry and algebra, and was moving in the right direction when he helped to close the gap
between numerical and geometric algebra with his geometric solution of the general cubic
equations, but the decisive step came later with Descartes.
Analytic geometry has traditionally been attributed to René Descartes who made significant
progress with the methods of analytic geometry when in 1637 in the appendix entitled Geometry
of the titled Discourse on the Method of Rightly Conducting the Reason in the Search for Truth
in the Sciences, commonly referred to as Discourse on Method. This work, written in his native
French tongue, and its philosophical principles, provided the foundation for calculus in Europe.
Abraham de Moivre also pioneered the development of analytic geometry. With the assumption
of the Cantor-Dedekind axiom, essentially that Euclidean geometry is interpretable in the
language of analytic geometry (that is, every theorem of one is a theorem of the other), Alfred
Tarski's proof of the decidability of the ordered real field could be seen as a proof that Euclidean
geometry is consistent and decidable
Introduction to Coordinate Geometry
from Latin: coordinate "to set in order, arrange"
Coordinate geometry is that branch of geometry in which two real numbers called coordinates
are used to indicate the position of point in a plane. The main contribution of coordinate
geometry is that it has the fact that it has inabled the integration of alzebra and geometry. This is
an evident from the fact that alzebric methods are employed to represent and prove the
fundamental properties of geometrical theorems. Equations are also employed to represent the
various geometric figures. It is because of these features that the coordinate geometry is
considerd to be most powerful tool to analysis than the Euclidian geometry. It is on this
consideration that sometimes it describes as analytical geometry.
Directed lines:- a directed line is a straight line with number units positive , zero and
negative. The point of origin is number 0. The arrow indicates its direction . on the side of arrow
are the positive numbers and on the other side are the negative numbers. It is like a real number
scale illustrated below:
A directed line can be horizontal normally indicated by XOX’ axis and vertical indicated
normally by YOY’ axis. The point where the two intersects is called point of origin. The two
lines together are called rectangular axis and are at right angle to each other. If these axes are noy
at right angles they are said to be oblique axis and the angle between the positive axes XOY is
denoted by ω (omega).
What are coordinates?
To introduce the idea, consider the grid on the right. The
columns of the grid are lettered A,B,C etc. The rows are numbered 1,2,3 etc from the top. We
can see that the X is in box D3; that is, column D, row 3.
D and 3 are called the coordinates of the box. It has two parts: the row and the column. There are
many boxes in each row and many boxes in each column. But by having both we can find one
single box, where the row and column intersect.
The Coordinate Plane
In coordinate geometry, points are placed on the "coordinate plane" as shown below. It has two
scales - one running across the plane called the "x axis" and another a right angles to it called the
y-axis. (These can be thought of as similar to the column and row in the paragraph above.) The
point where the axes cross is called the origin and is where both x and y are zero.
On the x-axis, values to the right are positive and those to the left are negative.
On the y-axis, values above the origin are positive and those below are negative.
A point's location on the plane is given by two numbers, one that tells where it is on the x-axis
and another which tells where it is on the y-axis. Together, they define a single, unique position
on the plane. So in the diagram above, the point A has an x value of 20 and a y value of 15.
These are the coordinates of the point A, sometimes referred to as its "rectangular
For a more in-depth explanation of the coordinate plane see The Coordinate Plane.
For more on the coordinates of a point see Coordinates of a Point
The coordinate plane is a two-dimensional surface on which we can plot points, lines and curves.
It has two scales, called the x-axis and y-axis, at right angles to each other. The plural of axis is
'axes' (pronounced "AXE-ease").
The horizontal scale is called the x-axis and is usually drawn with the zero point in the middle. As you go
to the right on the scale the values get larger and are positive. As you go to the left, they get larger but
The vertical scale is called the y-axis and is also usually drawn with the zero point in the middle. As you
go up from zero the numbers are increasing in a positive direction. As you go down from zero they
increase but are negative.
The point where the two axes cross (at zero on both scales) is called the origin. In the figure above you
can drag the origin point to reposition it to a more suitable location at any time.
When the origin is in the center of the plane, they divide it into four areas called
quadrants. The first quadrant, by convention, is the top right, and then they go around counter-
clockwise. In the diagram above they are labelled Quadrant 1,2 etc. It is conventional to label
them with numerals but we talk about them as "first, second, third, and fourth quadrant".
In the diagram above, you can drag the origin all the way into any corner and display just one
quadrant at a time if you wish.
Things you can do in Coordinate Geometry
If you know the coordinates of a group of points you can:
• Determine the distance between them
• Find the midpoint, slope and equation of a line segment
• Determine if lines are parallel or perpendicular
• Find the area and perimeter of a polygon defined by the points
• Transform a shape by moving, rotating and reflecting it.
• Define the equations of curves, circles and ellipses.
Information on all these and more can be found in the pages listed below.
Coordinates of midpoint:- we can find out the coordinates of a mid-point from
the coordinates of the any two points using the following formula :
Xm = X1+X2 , Ym = Y1+Y2
For example, the coordinates of the midpoint of the join of points
(-2,5),(6,3) are ( -2+6/2 , 5+3/2 ) , ie ., (2,4)
This is helpful first in finding out the middle point from the joint of any two points and secondly
in verifying whether two straight lines bisect each other.
In the diagram , the dotted vertical lines are drawn perpendicular to x-axis and the dotted horizontal line
are parallel to the x-axis. The ΔNMP and ΔQml are the congruent triangles. It follows , therefore , that
(Xm-X1) = (X2-Xm)
Xm = (X1+X2)/2 …………………….(1)
Also from the same congruent triangles, we get
Ym=(Y1+ Y2)/2 ……………………………….(2)
From equation 1 and 2 , we coclude that the coordinates of the midpoint (Xm,Ym) are
( (X1+X2)/2 , (Y1+Y2)/2 ) .
Distance between two points :-
The distance , say d, between two points P(x1,y1) and Q(x2,y2) is given by the formula
d= √ (x2-x1)² + (y2-y1)²
since we take square of the two diffrences , we may get designate any mof the points as (x1,x2) and the
In order to prove above formula , let us take two points P(x1,y1) and Q(x2,y2) as shown in following
The vertical dotted lines PB and QC are perpendicular from P and Q on the X axis and PR is the
perpendicular from P and Qc. Then
From the right angled triangle PRQ, right angled at R, we have by Pythagoras theorem
d2 = (x2-x1)2+(y2-y1)2
d = √(x2-x1)2+(y2-y1)2
The coordinates of point R(x,y) dividing the line in the ratio of m:n connecting the points P(x1,y1) and Q(x2
, y2) in the diagram below
x= and y=
Draw PL, RM and QN perpendiculars on XOX’ and draw PK and RT perpendiculars on RM and QN
We are given
Suppose PQ makes angle Ѳ with the X-axis .from the figure:
PR cosѲ =x-x1
and in ∆RQT
RQ cos Ѳ =x2-x
Dividing (1) by (2) we get
it may be noted that m which corresponds to the segment PR multiplies the coordinate of Q, while n,
which refers to RQ, multiplies the coordinate of P. if m=n we find that the coordinate of the midpoint
In the above it is assumed that the point R divides PQ internally in the ratio m: n. This means that the
point R lies between P and Q. if PQ is divided externally by R, then R lies outside PQ. The student should
repeat the above method and using the same diagram with R and Q interchanged, it can be proved that
Coordinates of centroid
The centroid of a triangle is the point of intersection of the three medians of triangle. Each median
bisects the side opposite to the vertex into two equal parts. In order to prove that the median of triangle
intersects at a point called centroid, we have to showthat the coordinates are
X= and y=
To prove this let us take a triangle with its vertices A(x1,y1), B(x2,y2) and C(x3,y3) as shown in the
In the diagram median AD bisects the base BC at D with coordinates
we take a point G where no medians intersect which divides AD internally , say, in ratio 2:1
,i.e.,m1:m2 = 2:1
hence by the section formula the coordinates of G are
Area of triangle
We can find out the area of the triangle with the vertices given.
For this let A(x1,y1), B(x2,y2) and C(x3,y3) be the coordinates of the vertices of the triangle ABC.
From A,B,C draw the perpendiculars AL, BM and CN on the x-axis.
As is clear from the figure area of ∆ABC
= area of trapezium ABML+area of trapezium ALNC – area of trapezium BMNC
Since the area of the trapezium = [sum of the parallel sides] × [perpendicular distance
between the parallel sides]
Hence the area of the ∆ABC can be given as
∆ABC = [BM+AL]ML + (AL+CN)LN - (BM+CN)MN
∆ABC = ]
Note: - if the three points are collinear then the area of ∆ABC is equal to zero
The straight line
The study of curve starts with the straight line which is simplest geometrical entity.
Mathematically it is defined as the shortest distance between the two points.
Slope of a straight line
Slope of the line is the tangent of the angle formed by the line above the x-axis towards its
positive direction, whatever be the position of the line as shown below :
Slope of a line is generally denoted by m . thus if a line makes an angle Ѳ with the positive
direction of the x-axis , its slope is
if Ѳ is acute , slope is positive and if Ѳ is obtuse , the slope is negative .
in terms of the coordinates , the slope of the line joining two points , say A(x1,y1) and B(x2,y2)
is given by:
The following diagrams will make the explanation more clear
DIFFERENT FORMS OF EQUATIONS OF THE STRAIGHT LINE
1.Equations of coordinate axes:
If P(x,y) be the point on the x-axis , then its ordinate y is always zero for any position of the
point P on the x-axis and for no other point.
Is the equation of x axis.
If P(x,y) is any point on the y-axis , then its abscissa x is always zero for any position of the point
P on the y-axis and for no other point.
... x=0, is the equation of y-axis.
Equation of lines parallel to the coordinate axis:-
Let P(x,y) be any point on a line parallel to y-axis at a distance ‘a’ from it . for any position of the
point P lying on this line and for no other point the abscissa x is always constant and is equal to
... x=a is the equation of the line parallel to the y-axis and at a distance a from it.
Similarly y=b is the equation of the line parallel to the x-axis and at a distance b from it .
Origin slope form
The eq. of line passing through the origin and having slope m:
Let a straight line pass through the origin o and have a slope m. let P(x,y) br any point on the
line. From P draw PM perpendicular on the x-axis then,
which is the required eq. of the line.
A line intercepting the axes :-
In case the straight line meets the x-axis and the y-axis at points other than the origin, the
respective points will be called x-intercept and y-intercept. The diagram shows the two
intercepts of a straight line AB which meet the x-axis in A and the y-axis in B, the OA is called
the intercept of the line on the y-axis , and the two intercept OA and OB taken in the particular
order are called the intercepts of the line on the axes.
It may be noticed that at the point of y intercept , the value of x is equal to zero and at the
point of x-intercept, the value of y is equal to zero. Therefore in order to find out the value of
say y-intercept we have to put x equal to zero in the equation. Similarly to find out the value of
x-intercept , we put y equal to zero in the equation.
Slope intercept form:-
The eq. of line with the slope m and an intercept c on y-axis.
Let a straight line of slope m intersects the y axis in K. let OK the intercept on the y-axis , be c .
then the coordinates of K are (0,c).
Take P(x,y) any variable point on the line.
... Slope of KP = …………(1)
Slope of line as given in m …………….(2)
Equating eq. (1) and (2) , we get the required equation as
In the figure if the point P(x,y) is taken at a distance r from the point r from the point R on the
line , then
... is the required eq. of the straight line in parametric form.
Two point form:-
The eq. of straight line passing through two points (x1,y1) and (x2,y2)
Let Q(x1,y1) and R(x2,y2) be the two points on the line. Take any point P(x,y) on the line.
Then by def.,
Slope of QP= ………..(1)
Slope of QR= …………(2)
Since A,P,B are collinear points from (1) and (2) , we have
Hence the required fro straight line is
Normal perpendicular form:-
The eq of straight line in terms of the perpendicular form the origin and the inclination of the
perpendicular with the axis:
Let a straight line be at a perpendicular distance p from the origin and the inclination of the
perpendicular OM with the x-axis be α. The coordinates of M , the foot of the perpendicular , in
terms of the given constants are (p cos α , p sin α ).
The inclination of the line with the x-axis is
Slope of the given line = tan
let P(x,y) be any point on the line , then
slope of MP=
since A,P,B are collinear from(1) and (2), we have
is the required equation.
General equation of a straight line
An equation of the form ax+by+c=0, where a, b, c are constants and x, y are variables, is called
the general equation of the straight line.
Slope of the line ax+by+c=o
The angle between the two straight lines
Let AB and CD be two straight lines with given inclination to the x-axis as Ѳ1 and Ѳ2, form
interior and exterior angles Ѳ and α respectively as shown below:
... α=∏-Ѳ ………(2)
1. For the interior angle:
if we express the slopes in terms of m1 and m2 , then the formula can also be expressed
2.For exterior angle, therefore
tanα=tan(∏-Ѳ) = -tan Ѳ = -
Condition Of Parallelism
If two line are parallel, the angle between them is zero degree.
m1-m2=0, i.e., m1=m2
Condition Of Perpendicularism
If the two lines are perpendicular
Length of perpendicular from a point
Let AB be the line
let P(x1,y1) be the points and PM perpendicular to AB. Join AP and PB .
AB meets the x-axis , where putting y=0 in (1), we get
ax+c=0 or x=-
... coordinate of A is
Similarly coordinate of B is (0,-
So the length of perpendicular=
Uses of coordinate geometry in daily life
In construction of buildings
In determining length between two
In finding slope of any thing.
Business mathematics (S.D.Sharma)