Rectangular Axis
Coordinate Geometry
The branch of mathematics in which we study the position of any
object lying in a plane with the help of two mutually perpendicular
lines in the same plane, is called coordinate geometry.
Rectangular Axis
Let XOX ′ and YOY ′ be two fixed straight lines, which meet at right
angles at O. Then,
Y
P (x, y)
X' X
O
Y'
(i) X ′OX is called axis of X or abscissa or the X-axis.
(ii) Y ′ OY is called axis of Y or ordinate or the Y-axis.
(iii) The ordered pair of real numbers ( x , y ) is called cartesian
coordinate.
(iv) Coordinates of the origin are ( 0, 0).
(v) y-coordinate of a point on X-axis is zero.
(vi) x-coordinate of a point on Y -axis is zero.
, Quadrants
The X and Y-axes divide the coordinate plane into four parts, each part
is called a quadrant which is given below
Y
(–, +) (+, +)
II Quadrant I Quadrant
x < 0, y > 0 x > 0, y > 0
X' X
(–, –) (+, –)
III Quadrant IV Quadrant
x < 0, y < 0 x > 0, y < 0
Y'
Polar Coordinates
In ∆OPQ,
Y
P (x, y)
r
y
θ
X′ X
O x Q
Y′
x y
cosθ = and sinθ = ⇒ x = r cosθ and y = r sinθ
r r
y
where, r = x 2 + y 2 and θ = tan−1
x
The polar coordinate is represented by the symbol P (r , θ ).
Distance Formulae
(i) Distance between two points P ( x1 , y1 ) and Q ( x2 , y2 ), is
PQ = ( x2 − x1 )2 + ( y2 − y1 )2 .
P (x1, y1) Q (x2, y2)
(ii) If points are (r1 , θ1 ) and (r2 , θ 2 ), then distance between them is
r12 + r22 − 2 r1r2 cos(θ1 − θ 2 ).
Coordinate Geometry
The branch of mathematics in which we study the position of any
object lying in a plane with the help of two mutually perpendicular
lines in the same plane, is called coordinate geometry.
Rectangular Axis
Let XOX ′ and YOY ′ be two fixed straight lines, which meet at right
angles at O. Then,
Y
P (x, y)
X' X
O
Y'
(i) X ′OX is called axis of X or abscissa or the X-axis.
(ii) Y ′ OY is called axis of Y or ordinate or the Y-axis.
(iii) The ordered pair of real numbers ( x , y ) is called cartesian
coordinate.
(iv) Coordinates of the origin are ( 0, 0).
(v) y-coordinate of a point on X-axis is zero.
(vi) x-coordinate of a point on Y -axis is zero.
, Quadrants
The X and Y-axes divide the coordinate plane into four parts, each part
is called a quadrant which is given below
Y
(–, +) (+, +)
II Quadrant I Quadrant
x < 0, y > 0 x > 0, y > 0
X' X
(–, –) (+, –)
III Quadrant IV Quadrant
x < 0, y < 0 x > 0, y < 0
Y'
Polar Coordinates
In ∆OPQ,
Y
P (x, y)
r
y
θ
X′ X
O x Q
Y′
x y
cosθ = and sinθ = ⇒ x = r cosθ and y = r sinθ
r r
y
where, r = x 2 + y 2 and θ = tan−1
x
The polar coordinate is represented by the symbol P (r , θ ).
Distance Formulae
(i) Distance between two points P ( x1 , y1 ) and Q ( x2 , y2 ), is
PQ = ( x2 − x1 )2 + ( y2 − y1 )2 .
P (x1, y1) Q (x2, y2)
(ii) If points are (r1 , θ1 ) and (r2 , θ 2 ), then distance between them is
r12 + r22 − 2 r1r2 cos(θ1 − θ 2 ).
