Ordinate

Coordinate geometry uses two primary concepts to locate points precisely in a two-dimensional plane: the ordinate (y-coordinate) and the abscissa (x-coordinate). These coordinates are fundamental for understanding geometric figures, graph plotting, and solving practical problems in various fields, such as mathematics, engineering, and physics.

Ordinate Meaning

The ordinate refers to the vertical distance or the y-coordinate of a point in the Cartesian coordinate system. It measures how far up or down a point is located from the origin (0,0). For any point represented as (x, y), the ordinate is the y-value.

Ordinate and Abscissa Explained

In coordinate geometry, any point on a plane is represented as (x, y), where:

• Abscissa is the x-coordinate, measuring horizontal distance.
• Ordinate is the y-coordinate, measuring vertical distance.

Example: For point (5, 3):

• Abscissa (x-coordinate) = 5
• Ordinate (y-coordinate) = 3

Ordinate Meaning in Hindi

In Hindi, ordinate is called निर्देशांक or अनुप्रस्थ निर्देश, representing the vertical location of a point:

Points on the Coordinate Axes

Ordinate of Points on the Y-axis

• Points on the Y-axis have the abscissa (x) equal to 0.
• Ordinate (y) can be any real number (positive, negative, or zero).
• Example points: (0, 5), (0, -2), (0, 0).

Ordinate of Points on the X-axis

• Points on the X-axis have the ordinate (y) equal to 0.
• Abscissa (x) can be any real number.
• Example points: (4, 0), (-6, 0), (0, 0).

Ordinate and Quadrants

The Cartesian plane is divided into four quadrants, each determining the sign of the ordinate and abscissa:

Practical Applications of Ordinate and Abscissa

Understanding ordinate and abscissa is crucial in:

• Graph plotting and data analysis.
• Physics for determining motion trajectories and forces.
• Engineering for designing mechanical parts and structures.
• Computer graphics and digital imaging.
• Navigation and geographic information systems (GIS).

Cartesian Plane

The Cartesian plane, named after French mathematician René Descartes, is a two-dimensional coordinate system formed by two perpendicular number lines. The horizontal line is called the x-axis, and the vertical line is called the y-axis. These axes divide the plane into four quadrants and provide a framework for locating points using ordered pairs (x,y). The Cartesian plane serves as the foundation for coordinate geometry, allowing mathematical relationships to be visualized graphically.

Origin

The origin is the point where the x-axis and y-axis intersect, denoted as (0,0). It serves as the reference point for measuring all coordinates in the plane. Distances to the right of the origin along the x-axis are positive, while distances to the left are negative. Similarly, distances above the origin along the y-axis are positive, while distances below are negative. The origin is the central point from which all measurements in the coordinate system begin.

Coordinates

Coordinates are ordered pairs (x,y) that uniquely identify the position of a point in the Cartesian plane. The first number (x) represents the horizontal distance from the origin, while the second number (y) represents the vertical distance. Coordinates allow us to translate geometric problems into algebraic form and vice versa. For example:

• Point (3,4) is located 3 units to the right and 4 units up from the origin
• Point (-2,5) is located 2 units to the left and 5 units up from the origin
• Point (4,-3) is located 4 units to the right and 3 units down from the origin
• Point (-5,-6) is located 5 units to the left and 6 units down from the origin

Quadrants

The x-axis and y-axis divide the Cartesian plane into four regions called quadrants, each with distinct properties regarding the signs of coordinates:

• First Quadrant (I): Contains points where both x and y are positive (+,+). Located in the upper right section of the plane.
• Second Quadrant (II): Contains points where x is negative and y is positive (-,+). Located in the upper left section of the plane.
• Third Quadrant (III): Contains points where both x and y are negative (-,-). Located in the lower left section of the plane.
• Fourth Quadrant (IV): Contains points where x is positive and y is negative (+,-). Located in the lower right section of the plane.

Understanding quadrants helps in quickly determining the general location of a point based on the signs of its coordinates.

Distance Formula

The distance formula calculates the straight-line distance between two points in the Cartesian plane. For points (x₁,y₁) and (x₂,y₂), the distance is: d = √[(x₂-x₁)² + (y₂-y₁)²]

This formula is derived from the Pythagorean theorem and is essential for measuring distances in coordinate geometry.

Slope

The slope of a line measures its steepness and direction. It is calculated as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. For two points (x₁,y₁) and (x₂,y₂), the slope is: m = (y₂-y₁)/(x₂-x₁)

The slope is positive if the line rises from left to right, negative if it falls from left to right, zero if it's horizontal, and undefined if it's vertical.

Midpoint

The midpoint of a line segment is the point that lies exactly halfway between the endpoints. For two points (x₁,y₁) and (x₂,y₂), the midpoint is: M = ((x₁+x₂)/2, (y₁+y₂)/2)

The midpoint formula is useful in various geometric calculations and constructions.

Coordinate Axes

The coordinate axes are the perpendicular reference lines that form the Cartesian plane:

• X-axis: The horizontal reference line where y = 0
• Y-axis: The vertical reference line where x = 0

Points that lie exactly on these axes have special properties:

• Points on the x-axis have y-coordinate equal to 0
• Points on the y-axis have x-coordinate equal to 0
• Only the origin (0,0) lies on both axes simultaneously

Section Formula

The section formula is used to find the coordinates of a point that divides a line segment in a given ratio. If a point P divides the line segment joining points A(x₁,y₁) and B(x₂,y₂) in the ratio m:n, then the coordinates of P are: P = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n))

This formula is particularly useful in problems involving internal and external division of line segments.

Polar Coordinates

While the Cartesian system uses perpendicular axes, polar coordinates specify a point using:

• r: The distance from the origin to the point
• θ: The angle formed with the positive x-axis

The relationship between Cartesian (x,y) and polar (r,θ) coordinates is:

• x = r·cos(θ)
• y = r·sin(θ)
• r = √(x² + y²)
• θ = tan⁻¹(y/x)

Polar coordinates are often more convenient for describing circular paths and periodic phenomena.