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| Coordinate
geometry or Analytic geometry |
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Line in a coordinate plane
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Hessian normal form of the equation of a line |
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Distance between a point and a line |
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Lines and
points relations, examples
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Hessian normal form of the equation of a line
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The normal p
from the origin to a line subtends the angle j
with the x-axis
and form two right triangles with
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axes as is shown in the figure below,
then
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| Hessian
normal form of the equation of a line. |
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By comparing the normal form with the
general form Ax
+ By + C = 0, of the
equation of the line, we can
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find relations between corresponding
coefficients of both forms so, if we write
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lAx
+ lBy
+ lC
= 0
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and after comparing both
equations, lA
= cosj
and lB
= sinj
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then, by plugging into the general
form obtained is
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the normal form of the equation of a
line expressed by coefficients of the general form, where the sign of the
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square root should be opposite to the
sign of
C.
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Distance between a point and a line
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The distance from a point
A(x0,
y0) to a line
l, equals the distance between the given line and a line passing
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through the given point parallel to
the given line, as shows the figure below.
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So, the length of the normal from the origin
to that new line is p +
d, and the coordinates of
A will then
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satisfy equation of the line, that is
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x0 cos
j
+ y0 sin
j
= p +
d or |
| d
= x0 cos
j
+ y0 sin
j
-
p. |
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If equation of a line is given in the general form, then
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| The sign of the square root is taken to be opposite to
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| sign of
C, and
where the negative result
shows that
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the same side as the origin in
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Example:
Given is a point A(-2, 3) and the direction vector of a line
s = i
-
2j, determine equation of the
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line through the point A
and draw a diagram.
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Solution:
Plug the point A
and the components of the direction vector into the parametric equation of a line:
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Example:
Write equation of a line in the point slope form and in parametric form if the line passes through
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the point A(2,
-1) and whose slope
m = -3, draw a diagram.
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Solution:
Plug the given elements of the line into equation
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Example:
A line passes through points
A(-3,
-4) and
B(9,
4). Find the general form of its equation and
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the segments that the line cuts on the coordinate axes.
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Solution:
From the equation of a line through two points
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Example:
Check if points, A(2,
-1),
B(5, -5) and
C(-1,
3) all lie on the same line.
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Solution:
If given points lie on the same line then must be satisfied following condition
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Check can also be done by examining if given points satisfy
vector’s equation
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by taking coordinates of any of three given points as the coordinates of the radius vector r.
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Example:
A line through a point A(-3,
5) forms a triangle with the coordinate axes whose area is 24
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square units, find equation of the line.
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Area of a triangle |
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Coordinates of the point
A
must satisfy equation of the line
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The solution n
= 4 satisfies given conditions, so that
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Example:
Find the distance of the line -4x
+ 3y -
25 = 0 from the origin.
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Solution:
Rewrite the equation of given line from the general form to the normal form,
that is
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Comparing with the normal form xcosj +
ysinj
-
p = 0, follows that the distance
p = 5.
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| Coordinate
geometry contents |
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