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| Coordinate
geometry or Analytic geometry |
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Line in a coordinate plane
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Slope-intercept form of a line,
y = mx + c |
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Slope or gradient, y-intercept and x-intercept
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The
intercept form of the equation of the line
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Lines parallel to the coordinate axes, horizontal and
vertical lines |
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The
point slope form of the equation of a line
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The
two point form of the equation of a line |
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Line in a coordinate plane
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| Slope-intercept form
of the equation of a line |
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m
is the slope, |
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x0 is
the
x-intercept, |
| c
is the
y-intercept. |
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| The
points at which the line y
= mx + c intersects the coordinate
axes are
denoted as, x0
and c,
in the picture above. |
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The
intercept form of the equation of a line
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We use the slope intercept form
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to get the intercept form of the
equation of the line (see the figure above).
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| Lines
parallel to the axes, horizontal and vertical lines
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| If
m =
0,
to every number
x
associated is the same constant value y =
c.
A
line parallel to the
x-axis
is called a horizontal line (or constant). |
| If
the x
value never changes a line is parallel to the y-axis.
A line parallel to the y-axis
is called a vertical line. |
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| The
point slope form of the equation of a line |
| The
equation of a line that passes through the given point (x1,
y1)
and has the given slope m
is represented by the
definition of the slope and is called point-slope 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 |
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| or
y
-
y1
= m(x -
x1) |
| The
equation can also be considered as |
| y
= mx translated to the
point P1(x1,
y1). |
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| Example:
Find the equation of the line that is parallel with the line y
= - x
- 2 and passes through the point
P1( 2,
1) . |
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| The two point form of
the equation of a line |
| Two
points P1(x1,
y1)
and P2(x2,
y2)
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 P1(x1,
y1)
is y -
y1
= m(x
- x1).
As the point P2(x2,
y2)
lies on the same line, its coordinates must satisfy the same
equation, so y2
- y1
= m(x2
- x1). |
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| Thus,
the slope |
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| then |
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the equation of the line passing through the two
points. |
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| Example:
Find the equation of the line which passes through points P(-2,
3) and Q(6,
-1). |
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| Coordinate
geometry contents |
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