COORDINATE GEOMETRY-Notes

Coordinate geometry (also called Cartesian geometry) is the branch of mathematics that helps us describe and study the position of points, lines, and shapes on a plane using numbers called coordinates. It combines the ideas of geometry (shapes and figures) with algebraic equations and numbers.
This study was initially developed by the French philosopher and mathematician René Déscartes.

In the study of geometry, we have previously encountered the concept of a number line, where numbers are represented on a straight line. Extending this idea to two dimensions, we can represent the position of a point on a plane. This plane is divided by two mutually perpendicular lines known as the axes, and the system used for this representation is called the Cartesian Coordinate System.

The horizontal axis is referred to as the x-axis (denoted by \(XX′\)), while the vertical axis is referred to as the y-axis (denoted by \(YY′\)). These two axes intersect at a point known as the origin (O). The segments \(OX\) and \(OY\) represent the positive directions of the x-axis and y-axis, respectively, whereas \(OX′\) and \(OY'\) represent their corresponding negative directions.

The intersection of the axes divides the plane into four regions, known as quadrants, which are numbered I, II, III, and IV in an anticlockwise direction. This plane is termed the Cartesian Plane or xy-plane, and the two perpendicular lines forming it are collectively referred to as the Coordinate Axes.

Properties of Quadrants

• Quadrant I: \(x\) is positive, \(y\) is positive
• Quadrant II: \(x\) is negative, \(y\) is positive
• Quadrant III: \(x \)is negative, \(y\) is negative
• Quadrant IV: \(x\) is positive, \(y\) is negative

The position of every point on the plane is written as an ordered pair \((x, y)\):

• \(x-coordinate\):
  Abscissa is the term used in coordinate geometry to refer to the \(x-coordinate\) of a point in a Cartesian coordinate system. It represents the horizontal distance of the point from the y-axis. For example, in the point (3, 4), the abscissa is 3.
• \(y-coordinate\):
  Ordinate is the term used in coordinate geometry to refer to the y-coordinate of a point in a Cartesian coordinate system. It indicates the vertical distance of the point from the \(x-axis\). For example, in the point (3, 4), the ordinate is 4.

Example 1 : See the given Fig-1 below and complete the following statements:

1. The abscissa and the ordinate of the point B are _ _ _ and _ _ _, respectively. Hence, the coordinates of B are (_ _, _ _).
2. The x-coordinate and the y-coordinate of the point M are _ _ _ and _ _ _, respectively. Hence, the coordinates of M are (_ _, _ _).
3. The x-coordinate and the y-coordinate of the point L are _ _ _ and _ _ _, respectively. Hence, the coordinates of L are (_ _, _ _).
4. The x-coordinate and the y-coordinate of the point S are _ _ _ and _ _ _, respectively. Hence, the coordinates of S are (_ _, _ _).

Solution:

1. The abscissa and the ordinate of the point B are 4 and 3, respectively. Hence, the coordinates of B are (4, 3).
2. The x-coordinate and the y-coordinate of the point M are -3 and 4, respectively. Hence, the coordinates of M are (-3, 4).
3. The x-coordinate and the y-coordinate of the point L are -5 and -4, respectively. Hence, the coordinates of L are (-5, -4).
4. The x-coordinate and the y-coordinate of the point S are 3 and -4, respectively. Hence, the coordinates of S are (3, -4).

Example 2 : Write the coordinates of the points marked on the axes in Fig-2

Solution:

1. The Coordinates of A are: (4,0) as point A is 4 point away from \(y\)-axis and 0 point from \(x\)-axis. Therfore \(x\)-coodinate of A is 4 and \(y\)-coordinate of A is 0. Hence the coordinates of A are (4,0)
2. The Coordinates of B are: (0,3) as point B is 0 point away from \(y\)-axis and 3 point from \(x\)-axis. Therfore \(x\)-coodinate of B is 0 and \(y\)-coordinate of B is 3. Hence the coordinates of B are (0,3)
3. The Coordinates of C are: (-5,0) as point C is -5 point away from \(y\)-axis and 0 point from \(x\)-axis. Therfore \(x\)-coodinate of C is -5 and \(y\)-coordinate of C is 0. Hence the coordinates of C are (-5,0)
4. The Coordinates of D are: (0, -4) as point D is 0 point away from \(y\)-axis and -4 point from \(x\)-axis. Therfore \(x\)-coodinate of D is 0 and \(y\)-coordinate of D is -4. Hence the coordinates of D are (0,-4)
5. The Coordinates of E are: (\frac{2}{3}, 0) as point A is 4 point away from \(y\)-axis and 0 point from \(x\)-axis. Therfore \(x\)-coodinate of E is \(\frac{2}{3}\) and \(y\)-coordinate of E is 0. Hence the coordinates of E are \((\frac{2}{3},0)\)

Plot the points A(4,2), B(-5,3), C(-4,-5) and D(5,2)

Solution:

A(3,6), B(3,2)and C(8,2) are the vertices of a rectangle. Plot these points on graph paper and then use it to find the co-ordinates of vetex D

Solution:

Construction Procedure for Rectangle ABCD Given A(3,6), B(3,2), C(8,2)

Step 1: Draw the coordinate axes.
Draw two perpendicular lines intersecting at point O (the origin). Mark the x-axis (horizontal) and y-axis (vertical). Mark equal divisions along both axes; number appropriately (at least 0 to 9 on x, 0 to 7 on y).

Step 2: Plot given points.

• Point A(3,6): Start at the origin, move 3 units right (x=3) and 6 units up (y=6). Mark and label it A.
• Point B(3,2): Move 3 units right and 2 units up. Mark and label it B.
• Point C(8,2): Move 8 units right and 2 units up. Mark and label it C.

Step 3: Join the points.
Connect A to B, B to C with straight lines. These will form two sides of the rectangle.

Step 4: Locate point D.
Since a rectangle has opposite sides equal and parallel, point D must complete the rectangle.

Observe:
AB is vertical (x-coordinates the same: 3), BC is horizontal (y-coordinates the same: 2).
From point A(3,6), move horizontally to the right till your x-coordinate matches C's x-coordinate (which is 8), but keep y from A (which is 6). So, point D is at (8,6).

Step 5: Plot point D.
Move 8 units right and 6 units up. Mark and label as D.

Step 6: Complete the rectangle.
Join C to D and D to A with straight lines.

By plotting the following points on the same graph paper, check whetherthey are collinear or not
i. (3,5), (1,1) and (0,1)
ii. (-2,-1), (-1,-4) and (-4,1)

Construction:
1. Draw the axes:
Draw and label the x-axis and y-axis on graph paper, marking a suitable scale to include all given coordinates.

2. Plot the given points
For each set:
Mark each point by moving right/left for the x-value and up/down for the y-value, then label the points (e.g., A, B, C).
For (3,5), (1,1), (0,1): Plot each and label.
For (−2,−1), (−1,−4), (−4,1): Plot each and label.

3. Join the points
Draw straight lines connecting the points in each group (first through all three points of a group).

4. Check for collinearity
If all three points of a group lie on a straight line, they are collinear. If one does not lie on the line formed by the other two, they are not collinear.

Points:
i. (3,5), (1,1) and (0,1) are collinear, whereas
ii. (-2,-1), (-1,-4) and (-4,1) are not collinear

• To locate the position of an object or a point in a plane, we require two perpendicular lines. One of them is horizontal, and the other is vertical.
• The plane is called the Cartesian, or coordinate plane and the lines are called the coordinate axes.
• The horizontal line is called the \(x\)-axis, and the vertical line is called the \(y\)-axis.
• The coordinate axes divide the plane into four parts called quadrants.
• The point of intersection of the axes is called the origin.
• The distance of a point from the \(y\)-axis is called its \(x\)-coordinate, or abscissa, and the distance of the point from the \(x\)-axis is called its y-coordinate, or ordinate. If the abscissa of a point is \(x\) and the ordinate is \(y\), then \((x, y)\) are called the coordinates of the point.
• If the abscissa of a point is \(x\) and the ordinate is \(y\), then \((x, y)\) are called the coordinates of the point.
• The coordinates of a point on the \(x\)-axis are of the form \((x, 0)\) and that of the point on the \(y\)-axis are \((0, y)\).
• The coordinates of the origin are (0, 0).
• The coordinates of a point are of the form (+ , +) in the first quadrant, (–, +) in the second quadrant, (–, –) in the third quadrant and (+, –) in the fourth quadrant, where + denotes a positive real number and – denotes a negative real number.
• If \(x \ne y\), then \((x, y) \ne (y, x)\), and \((x, y) = (y, x),\) if \(x = y\).