Coordinate Geometry

To fix the position of a point on a single number line, one number is enough. But to fix the position of a point anywhere in a flat surface (a plane), one number alone is not enough — two are needed.

Take two number lines, one horizontal and one vertical, and place them so that they cross each other at right angles, at the point marked 0 on each. This crossing point is called the origin, and the plane formed by these two number lines is called the Cartesian plane (also called the coordinate plane or the xy-plane).

Worked example: Suppose a point P is 4 units to the right of the vertical line and 3 units above the horizontal line. Using only one number line, this position cannot be described — the horizontal distance (4) and the vertical distance (3) are both needed together, written as the ordered pair (4, 3), to fix P exactly.

The horizontal number line is called the x-axis and the vertical number line is called the y-axis. Every point in the plane can now be described uniquely by a pair of numbers, one for each axis.

On the Cartesian plane, the horizontal line is the x-axis and the vertical line is the y-axis. Numbers to the right of the origin on the x-axis are positive, and to the left are negative. Numbers above the origin on the y-axis are positive, and below are negative.

The position of any point P is written as an ordered pair (x, y):

• The first number, x, is called the abscissa of P — the perpendicular distance of P from the y-axis, measured along the x-axis.
• The second number, y, is called the ordinate of P — the perpendicular distance of P from the x-axis, measured along the y-axis.

Worked example: If a point P has coordinates (5, -2), then its abscissa is 5 (P is 5 units from the y-axis) and its ordinate is -2 (P is 2 units below the x-axis). We write this as P(5, -2).

The order of the two numbers matters: (5, -2) and (-2, 5) are different points. This is why the pair is called an ordered pair.

Plotting a point: To mark a point (a, b) on the plane using a suitable, consistent scale, start at the origin, move a units along the x-axis (right if a is positive, left if a is negative), then move b units parallel to the y-axis (up if b is positive, down if b is negative). For example, to plot A(3, -4): move 3 units right along the x-axis, then 4 units down parallel to the y-axis — A lands in Quadrant IV.

The x-axis and y-axis together divide the plane into four regions called quadrants. Starting from the region where both axes are positive and moving anticlockwise, they are labelled Quadrant I, II, III and IV.

• Quadrant I (top right): x > 0, y > 0 — both coordinates positive.
• Quadrant II (top left): x < 0, y > 0 — abscissa negative, ordinate positive.
• Quadrant III (bottom left): x < 0, y < 0 — both coordinates negative.
• Quadrant IV (bottom right): x > 0, y < 0 — abscissa positive, ordinate negative.

Worked example: In which quadrant does the point (-6, 2) lie? Here the abscissa is -6 (negative) and the ordinate is 2 (positive). Since x < 0 and y > 0, the point lies in Quadrant II.

A useful memory aid: read the signs as (x, y) and match them to the pattern (+, +), (-, +), (-, -), (+, -) for Quadrants I, II, III, IV respectively.

Not every point lies inside one of the four quadrants — some lie exactly on an axis.

• A point on the x-axis is not above or below the axis, so its ordinate (y-coordinate) is 0. Such a point has coordinates of the form (x, 0).
• A point on the y-axis is not to the left or right of the axis, so its abscissa (x-coordinate) is 0. Such a point has coordinates of the form (0, y).
• The origin lies on both axes at once, so both its coordinates are 0: O(0, 0).

Worked example: A point P lies on the x-axis, 7 units to the left of the origin. Since P is on the x-axis, its ordinate is 0; since it is 7 units in the negative direction, its abscissa is -7. So P = (-7, 0).

Worked example: A point Q lies on the y-axis, 3 units above the origin. Its abscissa is 0 (it is on the y-axis) and its ordinate is 3 (3 units up). So Q = (0, 3).

Points lying on an axis are not considered to be inside any quadrant — quadrants are the open regions strictly between the axes.

Once a point's position is fixed by its coordinates, a natural next question is: how far apart are two given points? The distance formula answers this using the Pythagoras theorem (known in India as the Baudhayana theorem, stated centuries before Pythagoras).

Let A(x1, y1) and B(x2, y2) be two points in the plane. Draw a line through A parallel to the x-axis and a line through B parallel to the y-axis; let them meet at point C. Then C has coordinates (x2, y1), and triangle ACB is right-angled at C.

• The horizontal leg AC = |x2 - x1|.
• The vertical leg BC = |y2 - y1|.

By the Pythagoras (Baudhayana) theorem, AB² = AC² + BC² = (x2 - x1)² + (y2 - y1)². Taking the square root of both sides gives the distance formula: AB = √[(x2 - x1)² + (y2 - y1)²].

Worked example: Find the distance between P(2, 3) and Q(5, 7). Here x1=2, y1=3, x2=5, y2=7. PQ = √[(5-2)² + (7-3)²] = √[3² + 4²] = √[9+16] = √25 = 5 units.

Worked example (distance from the origin): Find the distance of the point (6, 8) from the origin O(0, 0). Taking (x1,y1) = (0,0) and (x2,y2) = (6,8): OA = √[(6-0)² + (8-0)²] = √[36+64] = √100 = 10 units.