Analytical geometry
Introduction
The branch of mathematics where algebraic methods are employed for solving
problems in geometry is known as analytical geometry. It is sometimes called Cartesian
Geometry.
Let X’OX and Y’OY be two perpendicular straight lines intersecting at the point O.
The fixed point O is called origin. The horizontal line X’OX is known as X –axis and the
vertical line Y’OY be Y-axis. These two axes divide the entire plane into four parts known
as Quadrants.
Y
P ′ (-X,Y) P(X,Y)
X′ M O N X
Q ′ (-X,-Y) Q(X,-Y)
Y′
All the values right of the origin along the X-axis are positive and all the values
left of the origin along the X- axis are negative. Similarly all the values above the origin
along Y – axis are positive and below the origin are negative.
Let P be any point in the plane. Draw PN perpendicular to X –axis. ON and PN
are called X and Y co-ordinates of P respectively and is written as P (X,Y). In particular
the origin O has co-ordinates (0,0) and any point on the X-axis has its Y co-ordinate as
zero and any point on the Y-axis has its X-co-ordinates as zero.
Straight lines
A straight line is the minimum distance between any two points.
Slope
The slope of the line is the tangent of the angle made by the line with positive
direction of X – axis measured in the anticlockwise direction.
, B
Y
A θ
X′ O X
Y′
let the line AB makes an angle θ with the positive direction of X-axis as in the figure.
The angle θ is called the angle of inclination and tan θ is slope of the line or gradient of
the line. The slope of the line is denoted by m. i.e., slope = m = tan θ
Y Y
θ (obtuse) θ (acute)
X
X′
X
Y′
Y′
Slope = m = tan θ Slope = m= tan θ Slope is negative Slope is positive
Note
(i) The slope of any line parallel to X axis is zero.
(ii) Slope of any line parallel to Y axis is infinity
(iii) The slope of the line joining two points (x 1, y 1 ) and (x 2, y 2 ) is
y1 − y 2
Slope = m = tan θ =
x1 − x 2
Introduction
The branch of mathematics where algebraic methods are employed for solving
problems in geometry is known as analytical geometry. It is sometimes called Cartesian
Geometry.
Let X’OX and Y’OY be two perpendicular straight lines intersecting at the point O.
The fixed point O is called origin. The horizontal line X’OX is known as X –axis and the
vertical line Y’OY be Y-axis. These two axes divide the entire plane into four parts known
as Quadrants.
Y
P ′ (-X,Y) P(X,Y)
X′ M O N X
Q ′ (-X,-Y) Q(X,-Y)
Y′
All the values right of the origin along the X-axis are positive and all the values
left of the origin along the X- axis are negative. Similarly all the values above the origin
along Y – axis are positive and below the origin are negative.
Let P be any point in the plane. Draw PN perpendicular to X –axis. ON and PN
are called X and Y co-ordinates of P respectively and is written as P (X,Y). In particular
the origin O has co-ordinates (0,0) and any point on the X-axis has its Y co-ordinate as
zero and any point on the Y-axis has its X-co-ordinates as zero.
Straight lines
A straight line is the minimum distance between any two points.
Slope
The slope of the line is the tangent of the angle made by the line with positive
direction of X – axis measured in the anticlockwise direction.
, B
Y
A θ
X′ O X
Y′
let the line AB makes an angle θ with the positive direction of X-axis as in the figure.
The angle θ is called the angle of inclination and tan θ is slope of the line or gradient of
the line. The slope of the line is denoted by m. i.e., slope = m = tan θ
Y Y
θ (obtuse) θ (acute)
X
X′
X
Y′
Y′
Slope = m = tan θ Slope = m= tan θ Slope is negative Slope is positive
Note
(i) The slope of any line parallel to X axis is zero.
(ii) Slope of any line parallel to Y axis is infinity
(iii) The slope of the line joining two points (x 1, y 1 ) and (x 2, y 2 ) is
y1 − y 2
Slope = m = tan θ =
x1 − x 2
