Coordinate Geometry

Coordinate geometry, also known as analytical geometry, is a fascinating branch of mathematics that blends algebra with geometry. Unlike traditional geometry which deals mainly with the shape, size, and dimensions of objects such as points, lines, surfaces, and solids, coordinate geometry provides a connection between algebra and geometry through the graphs of equations. It uses a pair of numerical values called coordinates to determine the position of points, lines, and planes in two-dimensional space or even three-dimensional space.

Cartesian coordinate system

Named after French mathematician René Descartes, the Cartesian coordinate system is a two-dimensional plane defined by a horizontal number line (the x-axis) and a vertical number line (the y-axis). These axes intersect at a point called the origin, which is usually designated as the point (0,0). Each point in this plane is described by an ordered pair of numbers: the x-coordinate and the y-coordinate. The x-coordinate specifies the distance to the right or left of the y-axis, while the y-coordinate specifies the distance above or below the x-axis.

    x-axis ------------------ 
                           ,
                           | *(3, 4)
                           ,
                           ,
        -------------------|------------------- y-axis
                           ,
                           ,

In the diagram above, the star represents a point with coordinates (3, 4) in the two-dimensional plane. This means that the point is 3 units to the right of the y-axis and 4 units above the x-axis.

Plotting points

Plotting points in coordinate geometry is simple. You take an ordered pair and determine the position of the point belonging to that pair.

For example, let's plot the point (-2, 5):

    x-axis ------------------ 
           ,
           | * (-2, 5)
           ,
           ,
-------------------|------------------- y-axis
           ,       
           ,

Here, -2 means move 2 units to the left of the y-axis, and 5 means move 5 units up the x-axis, thus our point will come to the correct position.

Distance between two points

To find the distance between two points, you can use the distance formula derived from the Pythagorean Theorem. If you have two points, (x1, y1) and (x2, y2), the distance d between them is given by:

    D = √((x2 - x1)² + (y2 - y1)²)

For example, to find the distance between points A (1, 2) and B (4, 6):

    D = √((4 - 1)² + (6 - 2)²)
      = √(3² + 4²)
      = √(9 + 16)
      = √25
      = 5

The distance between the points is 5 units.

Midpoint of a line segment

The midpoint of a line segment connecting two points is simply the point dividing the line segment into two parts. Given the two endpoints, (x1, y1) and (x2, y2), the midpoint (xm, ym) is calculated as follows:

    xm = (x1 + x2) / 2
    ym = (y1 + y2) / 2

Let us find the midpoint of the line segment joining the points C (2, 3) and D (10, 9):

    xm = (2 + 10) / 2 = 12 / 2 = 6
    ym = (3 + 9) / 2 = 12 / 2 = 6

    Midpoint = (6, 6)

Slope of the line

The slope of a line is a measure of its steepness and direction. The formula for finding the slope m of a line through two points (x1, y1) and (x2, y2) is:

    m = (y2 - y1) / (x2 - x1)

Consider the points E (2, 3) and F (5, 7), the slope is:

    m = (7 - 3) / (5 - 2)
      = 4 / 3

The slope of the line joining points E and F is 4/3.

Equation of line

The equation of a line is described in several forms, the most common of which are slope-intercept form and point-slope form.

Slope-intercept form

This form of the equation for a line is expressed as y = mx + c, where m is the slope and c is the y-intercept (the point where the line crosses the y-axis).

For example, if the slope of a line is 2 and it intersects the y-axis at -3, the equation is:

    y = 2x – 3

Point-slope form

If you know a point (x1, y1) on the line and its slope m, you can write the equation using:

    y - y1 = m(x - x1)

Given a point G (3, 4) and slope 5, the equation is:

    y – 4 = 5(x – 3)

General form

The general form of a linear equation is expressed by Ax + By + C = 0, where A, B, and C are constants.

Converting point-slope or slope-intercept form to normal form involves algebraic manipulation, such as expanding parentheses and collecting like terms.

Parallel and perpendicular lines

Lines are parallel if they have the same slope. Thus, two lines with slopes m1 and m2 are parallel if m1 = m2.

For example, the lines y = 2x + 3 and y = 2x - 4 are parallel because they both have a slope of 2.

Lines are perpendicular if the product of their slopes is -1. Thus, lines with slopes m1 and m2 are perpendicular if m1 * m2 = -1.

For example, if one line is y = 3x + 5, then the other line is perpendicular if its slope is -1/3.

Conic sections

In coordinate geometry, the study also includes conic sections: circles, ellipses, parabolas, and hyperbolas. These are the curves obtained by intersecting a right circular cone with a plane.

Circle

The standard equation of a circle with centre (h, k) and radius r is:

    (x – h)² + (y – k)² = r²

The equation of a circle centered at (2, -1) and of radius 5 is:

    (x - 2)² + (y + 1)² = 25

Ellipse

The equation of the ellipse with center (h, k) is:

    (x - h)²/a² + (y - k)²/b² = 1

The values of a and b determine the shape and orientation of the ellipse.

Parabola

A parabola opening up/down with vertex at (h, k) is given by:

    (x – h)² = 4p(y – k)

where p is the distance from the vertex of the parabola to the focus.

Hyperbola

The standard equation is:

    (x - h)²/a² - (y - k)²/b² = 1

A hyperbola has two distinct curves called branches that are symmetrical to each other.

Conclusion

Coordinate geometry is an important subject and its concepts are applied in various fields such as physics, computer graphics, engineering, and navigation. Being able to represent geometric shapes with equations allows algebraic operations on these shapes to be performed more easily, which helps greatly in visualizing and solving problems that might otherwise be quite complex.

By understanding the basics of plotting points, calculating distances, determining slopes, and solving equations of lines, learners can solve more complex geometric problems more easily.

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