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Learning Analytical/Coordinate/Cartesian Geometry
...from page 8
Coordinates of a point on a line
Every line has a coordinate system. A line with a defined coordinate system is called the coordinate line. On a coordinate line every point can be associated with a unique real number called its coordinate. This coordinate would be either positive or negative or zero based on the location of the point with respect to the coordinate system defined on it.
Coordinate Plane formed by two coordinate lines
The coordinate plane is defined or formed by two mutually perpendicular coordinate lines. By convention we assume one of them to be horizontal (which we call the x-axis) and the other to be vertical (the y-axis).
Every point in the plane has two coordinates defining its location. The two coordinates are presented together in the form of an ordered pair.
Coordinates of a point in a Plane :: Projections of the point on the coordinate axes.
The coordinates of a point in a plane are defined with respect to their projections on the coordinate axes defining the plane as follows:
First Coordinate = The projection of the point on the horizontal axis
Second Coordinate = The projection of the point on the vertical axis.
| Eg: |
(a) |
Where "P" is a point in the First Quadrant of the coordinate plane and A (x) and B (y) are the projections of ‘P’ on the x – axis and y – axis respectively,
‘x’ is the first Coordinate (Or) x – coordinate (Or) abscissa of the point P
‘y’ is the second Coordinate (Or) y – coordinate (Or) ordinate of the point P
The ordered pair (x, y) i.e. the numbers x, y taken in that order are called the rectangular Cartesian coordinates or simply coordinates of P.
The point P is represented as P(x , y) or P = (x, y)
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Coordinates as Distances from the axes
The coordinates can also be defined/identified as the distance of the point from the axes.
The x-coordinate of a point indicates the distance of the point from the y-axis
The y-coordinate of a point indicates the distance of the point from the x-axis
| Eg: |
(a) |
For the point P,
The distance from x-axis is PB.
PB and OA are the opposite sides of the rectangle OAPB ⇒ PB = OA.
OA is the distance of ‘A’ from the origin ‘O’ on the coordinate line x'Ox ⇒ OA = x [The distance of a point on the coordinate line from the origin is the magnitude of its Coordinate)
Therefore PB = OA = x
The distance from y-axis is PA.
PA and OB are the opposite sides of the rectangle OAPB ⇒ PA = OB.
OB is the distance of ‘B’ from the origin ‘O’ on the coordinate line y'Oy ⇒ OB = y [The distance of a point on the coordinate line from the origin is the magnitude of its Coordinate)
Therefore PA = OB = y
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The point coordinates of any point in the plane can be defined/identified in this manner.
Sign of the Numerical Value of the Coordinates
The coordinates are positive or negative based on the location of the point.
If the point is located in the
- First Quadrant
The projection on the horizontal axis lies on the positive x-axis and on the vertical axis lies on the positive y-axis.
| Eg: |
For the Point "P" in the first quadrant
The projection on
- x-axis is "A(x)" which lies on the positive x-axis ⇒ x is positive.
- y-axis is "B(y)" which lies on the positive y-axis ⇒ y is positive..
The Coordinates of "P" = (x, y) [where x is +ve and y is +ve]
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- Second Quadrant
The projection on the horizontal axis lies on the negative x-axis and on the vertical axis lies on the positive y-axis.
| Eg: |
For the Point "H" in the second quadrant
The projection on
- x-axis is "N(a)" which lies on the negative x-axis ⇒ a is negative.
- y-axis is "M(b)" which lies on the positive y-axis ⇒ b is positive..
The Coordinates of "H" = (a, b) [where a is -ve and b is +ve]
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- Third Quadrant
The projection on the horizontal axis lies on the negative x-axis and on the vertical axis lies on the negative y-axis.
| Eg: |
For the Point "K" in the third quadrant
The projection on
- x-axis is "C(x1)" which lies on the negative x-axis ⇒ x1 is negative.
- y-axis is "D(y1)" which lies on the negative y-axis ⇒ y1 is negative.
The Coordinates of "K" = (x1, y1) [where x1 is -ve and y1 is -ve]
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- Fourth Quadrant
The projection on the horizontal axis lies on the positive x-axis and on the vertical axis lies on the negative y-axis.
| Eg: |
For the Point "F" in the third quadrant
The projection on
- x-axis is "S(xa)" which lies on the positive x-axis ⇒ xa is positive.
- y-axis is "T(ya)" which lies on the negative y-axis ⇒ ya is negative.
The Coordinates of "F" = (x1, y1) [where xa is +ve and ya is -ve]
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Coordinates of the Origin
The origin (Point "O") lies on both the axes. The coordinate of the origin with respect to the horizontal axis is O(0) and with respect to the vertical axis is also O(0).
The projection of a point on a line to which it belongs is the point itself.
⇒ The projection of "O" on
- x-axis is "O(0)"
- y-axis is "O(0)"
The Coordinates of the origin in the plane ⇒ O = (0,0)
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