Key facts

  • The x-axis is horizontal, the y-axis is vertical, and their intersection is the origin (0, 0).
  • A point with y = 0 lies on the x-axis, and a point with x = 0 lies on the y-axis.
  • The distance between A(x1, y1) and B(x2, y2) is sqrt((x2 - x1)^2 + (y2 - y1)^2).
  • Internal division in the ratio m:n uses ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)).
  • The midpoint is the 1:1 internal division point: ((x1 + x2)/2, (y1 + y2)/2).

Key Points at a Glance

  1. 1

    The x-axis is horizontal, the y-axis is vertical, and their intersection is the origin (0, 0).

  2. 2

    An ordered pair is always read as (x, y); changing the order gives a different point.

  3. 3

    Quadrant signs are I: (+, +), II: (-, +), III: (-, -), and IV: (+, -).

  4. 4

    A point with y = 0 lies on the x-axis, and a point with x = 0 lies on the y-axis.

  5. 5

    The distance between A(x1, y1) and B(x2, y2) is sqrt((x2 - x1)^2 + (y2 - y1)^2).

  6. 6

    For a horizontal segment, distance is the absolute difference of x-coordinates; for a vertical segment, it is the absolute difference of y-coordinates.

  7. 7

    Squared distance is a useful intermediate check before taking the positive square root.

  8. 8

    Internal division in the ratio m:n uses ((mx2 + nx1)/(m + n), (my2 + ny1)/(m + n)).

  9. 9

    The midpoint is the 1:1 internal division point: ((x1 + x2)/2, (y1 + y2)/2).

  10. 10

    External division in the ratio m:n uses ((mx2 - nx1)/(m - n), (my2 - ny1)/(m - n)).

  11. 11

    For external division, m = n gives no finite division point because the denominator becomes zero.

  12. 12

    Always write the ratio direction, such as AP:PB = m:n, before substituting in a section formula.

What are the basics of the coordinate plane?

The coordinate plane is a two-axis grid in which every point is located by an ordered pair (x, y), with the x-coordinate showing horizontal position and the y-coordinate showing vertical position. According to the Rajasthan Staff Selection Board, Jaipur official Clerk Grade-II/Junior Assistant Combined Direct Recruitment Advertisement 2024, the recruitment advertised 4,197 total posts, so this LDC-level mathematics topic belongs to a large competitive field where basic accuracy matters.

Cartesian coordinate geometry begins with a flat plane on which every point can be located by two numbers. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. They meet at the origin, written as (0, 0). The axes divide the plane into four regions called quadrants. In examination problems, most errors at this stage come from reading the ordered pair casually. The first number always belongs to the x-axis and the second number always belongs to the y-axis. Thus (3, -2) means move 3 units right from the origin and then 2 units down; it is not the same point as (-2, 3).

The sign of each coordinate tells the direction of movement from the origin. A positive x-coordinate means movement to the right, and a negative x-coordinate means movement to the left. A positive y-coordinate means movement upward, and a negative y-coordinate means movement downward. This gives the quadrant sign pattern: Quadrant I has (+, +), Quadrant II has (-, +), Quadrant III has (-, -), and Quadrant IV has (+, -). Points on the axes are not counted in any quadrant. If y = 0, the point lies on the x-axis; if x = 0, the point lies on the y-axis. The origin is the only point where both coordinates are zero.

Plotting a point is a two-step action. Start from the origin, follow the x-coordinate horizontally, then follow the y-coordinate vertically. For example, to plot A(-4, 5), move 4 units left and then 5 units up. To plot B(0, -3), do not move horizontally; move only 3 units downward on the y-axis. To plot C(6, 0), move 6 units right on the x-axis and stop there. This basic plotting sense is important because distance and section-formula questions become easier when the rough position of the points is clear.

A useful exam habit is to sketch only what is needed. A full graph paper drawing is rarely required in a numeracy test. Mark the axes, show the correct quadrant, and write the coordinates near the point. If a line segment joins two points in different quadrants, the signs already warn you whether subtraction will turn into addition in the formula. If both points have the same x-coordinate, the segment is vertical. If both have the same y-coordinate, the segment is horizontal. These visual checks prevent mistakes before formal calculation begins.

The coordinate plane also turns geometry into arithmetic. Lengths, midpoints, and division points can be found using the coordinates without drawing an exact figure. For the LDC syllabus, the core expectation is not advanced analytic geometry; it is confident handling of ordered pairs, signs, axes, distance, and division of a line segment in a given ratio. A candidate who can first locate the points correctly usually finds the formulas much easier to apply.