
Coordinate
geometry or Analytic geometry 


Line in a coordinate plane

Slopeintercept form of a line,
y = mx + c 
Slope or gradient, yintercept and xintercept

The
intercept form of the equation of the line

Lines parallel to the coordinate axes, horizontal and
vertical lines 
The
point slope form of the equation of a line

The
two point form of the equation of a line 





Line in a coordinate plane

Slopeintercept form
of the equation of a line 


m_{ }
is the slope, 
x_{0} is
the
xintercept, 
c
is the
yintercept. 


The
points at which the line y
= mx + c intersects the coordinate
axes are
denoted as, x_{0}
and c,
in the picture above. 


The
intercept form of the equation of a line

We use the slope intercept form


to get the intercept form of the
equation of the line (see the figure above).


Lines
parallel to the axes, horizontal and vertical lines

If
m =
0,
to every number
x
associated is the same constant value y =
c.
A
line parallel to the
xaxis
is called a horizontal line (or constant). 
If
the x
value never changes a line is parallel to the yaxis.
A line parallel to the yaxis
is called a vertical line. 



The
point slope form of the equation of a line 
The
equation of a line that passes through the given point (x_{1},
y_{1})
and has the given slope m
is represented by the
definition of the slope and is called pointslope form or the
gradient form of the line. 
Since the slope
of a line is the ratio of its vertical change to its horizontal
change then 



or
y

y_{1}
= m(x 
x_{1}) 
The
equation can also be considered as 
y
= mx translated to the
point P_{1}(x_{1},
y_{1}). 





Example:
Find the equation of the line that is parallel with the line y
=  x
 2 and passes through the point
P_{1}( 2,
1) . 


The two point form of
the equation of a line 
Two
points P_{1}(x_{1},
y_{1})
and P_{2}(x_{2},
y_{2})
determine a unique line on the Cartesian plane, therefore their
coordinates satisfy the equation y
= mx
+ c. 
The
equation of the line which passes through the point P_{1}(x_{1},
y_{1})
is y 
y_{1}
= m(x
 x_{1}).
As the point P_{2}(x_{2},
y_{2})
lies on the same line, its coordinates must satisfy the same
equation, so y_{2}
 y_{1}
= m(x_{2}
 x_{1}). 

Thus,
the slope 

then 

is
the equation of the line passing through the two
points. 




Example:
Find the equation of the line which passes through points P(2,
3) and Q(6,
1). 









Coordinate
geometry contents 



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