Popular Coordinate Geometry Notes

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CO-ORDINATE GEOMETRY
CARTESIAN CO-ORDINATE SYSTEM : B (x2 , y2)
Rectangular Co-ordinate System : Let X' OX
and Y'OY be two mutually perpendicular lines
through any point O in the plane of the pa- A (x1 , y1)
per. Point O is known as the origin. The line Note :
X'OX is called the x -axis or axis of x ; the 1. Distance is always positive. Therefore, we
line Y'OY is known as the y-axis or axis of y, often write AB instead of |AB|.
and the two lines taken together are called 2. The distance of a point P (x, y) from the ori-
the co-ordinates axes or the axes of co-ordi-
nates. gin  x 2  y 2
3. The distance between two polar co-ordinates
Y A (r1 , θ1 ) and B (r2, θ 2 ) is given by
3 AB  r12  r 2 2  2r1r2cos(θ1 – θ2 )
II 2 I
Quadrant Quadrant Application of Distance Formulae :
1 (i) For given three points A, B, C to decide
(-.,+) (+,+)
whether they are collinear or vertices of a
X’ X particular triangle. After finding AB, BC and
-3 -2 -1 0 CA we shall find that the points are :
-1 • Collinear - (a) If the sum of any two distances
III IV
Quadrant -2 Quadrant is equal to the third
(+,-) i.e. AB + BC = CA. or AB + CA = BC
(-,-) -3
or BC + CA = AB
(b) If are of  ABC is zero
Y’
(c) If slope of AB = slope of BC = slope of CA.
• Vertices of an equilateral triangle if AB = BC
= CA
• Vertices of an isosceles triangle if AB = BC
Signs of or BC = CA or CA = AB.
Quad- Nature of X
Region co-ordin- • Vertices of a right angled triangle if AB2 +
rant and Y
ate
BC2 = CA2 etc.
XOY I x > 0, y > 0 (+, +) (ii) For given four points A,B,C,D :
• AB = BC = CD = DA and AC = BD  ABCD is a
YOX' II x < 0, y > 0 (- , +) square.
X'OY' III x < 0, y < 0 (-, -) • AB = BC = CD = DA and AC  BD  ABCD is
a rhombus.
Y'OX IV x > 0, y < 0 (+, -) • AB = CD, BC = DA and AC = BD  ABCD is a
recatangle.
Note - Any point lying on x-axis or y-axis does • AB = CD, BC = DA and AC  BD  ABCD is a
not lie in any quadrant. parallelogram.
Any point can be represented on the plane Note :
described by the co-ordinate axes by specify- • The four given points are collinear, if Area of
ing its x and y co-ordinates. qaudrilateral ABCD is zero.
The x -co-ordinate of the point is also known
• Diagonals of square, rhombus, rectangle and
as the abscissa while the y-coordinate is also
parallelogram always bisect each-other.
known as the ordinate.
Distance Formula : The distance two point • Diagonals of rhombus and square bisect each
other at right angle.
A (x1, y1) and B(x2, y2) is given by
 Section Formuale :
AB  x 2  x1 2  y 2  y1 2 1.The co-ordinates of a point P(x ,y), dividing
the line segment joining the two points
A(x1,y1) and B (x 2, y2) internally in the ratio
m1 : m2 are given by



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m1x 2  m2 x1 m y  m2 y1  Some Important Points in a Triangle :
x ,y  1 2 • Centroid : If (x1, y1), (x2 , y2) and (x3, y3) are the
m1  m2 m1  m2
vertices of a triangle, then the co-ordinates
of its centroid are -
m2 B (x2 , y2)
AP m1  x1  x 2  x 3 y1  y 2  y 3 
 m1  , 
BP m 2 P(x, y)  3 3 
A (x1, y1) • Incentre : If A (x1, y1), B(x2, y2) and C(x3, y3)
2. The co-ordinate of the point P(x, y), dividing are the vertices of a  ABC s.t. BC = a, CA = b
the line segment joining the two points A(x1, and AB = c, then the co-ordinates of its
y1) and B (x 2, y2) externally in the ratio m1:m2 incentre are
are given by
m1x 2  m 2 x1 m y  m 2 y1 A (x1 ,y1)
x ,y  1 2
m1  m 2 m1  m 2

P (x, y)
AP m1
 B (x2, y2)
BP m 2 c b
A (x1 , y1)
3. The co-ordinates of the mid-point of the line
segment joining the two points A(x1, y1) and
B (x 2, y2) are given by
 x1  x 2 y1  y 2  B a C
 ,  (x 2 ,y2) (x3 ,y3)
 2 2 
B (x2, y2)
 ax1  bx 2  cx 3 ay1  by 2  cy 3 
 , 
P (x, y)  abc abc 
A (x1 , y1) • Circumcentre : If A(x1, y1), B (x2, y2) and C(x3
Division by Axes : If P (x1, x2) and Q (x2, y2), , y3) are the vertices of a  ABC , then the co-
then PQ is divided by ordinates of its circumcentre are
 y1  x1 sin2A  x 2 sin 2B  x 3 sin 2C
(i) x - axis in the ratio = y  ,
2  sin 2A  sin 2B  sin 2C

x1 y1 sin2A  y 2 sin 2B  y 3 sin 2C 
(ii) y - axis in the ratio =  x 
2 sin 2A  sin 2B  sin 2C 
Division by a Line : A line ax + by + c = 0 • Orthocentre : Co-ordinates of orthocentre are
ax1  by1  c  x1 tanA  x 2 tan B  x 3 tan C
divides PQ in the ratio =   ,
ax 2  by 2  c  tan A  tan B  tan C
 Area of a triangle : The area of a triangle y1 tanA  y 2 tan B  y 3 tan C 
ABC whose vertices are (x1, y1), B(x2, y2) and 
tan A  tan B  tan C 
C(x3 , y3) is denoted by  .
Note :
x1 y1 1 • If the traingle is equilateral, then centroid,
1
 Δ x2 y2 1 incentre, orthocentre, circumcentre coin-
2 x3 y3 1 cides.
• Orthocentre, centroid and circumcentre are
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 x1 y 2  y 3   x 2 y 3  y1   x 3 y1  y 2  always collinear and centroid divides the line
2 joining orthocentre and circumcentre in the
 Area of Polygon : The area of the polygon ratio 2 : 1.
whose vertices are (x1, y1), (x2, y2),......(xn , yn) • In an isosceles triangle centroid,
is - orthocentre, incentre, circumcentre lies on
1 x1 y 2  x 2 y1   x 2 y 3  x 3 y 2   ....... the same line.
  Incentre divides the angles bisectors in the
2 ................  x n y1  x1 y n   •
ratio (b + c) : a, (c + a) : b, (a + b) : c.


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