Definition and Basic Concepts

Coordinate geometry is based on the concept of a coordinate system, which represents points in a two-dimensional space using a set of numerical coordinates. The most common coordinate system used in coordinate geometry is the Cartesian coordinate system, which is named after the French mathematician René Descartes.

Cartesian Coordinate System

The Cartesian coordinate system, also known as the rectangular coordinate system, is in which a pair of numerical coordinates represent each point in a plane. These coordinates are represented by an ordered pair (x, y), where x is the horizontal coordinate and y is the vertical coordinate. The point where the x-axis and y-axis intersect is called the origin and is represented by the coordinates (0, 0).

The Distance Formula

The distance formula is used to calculate the distance between two points in a Cartesian coordinate system. The formula is given by

                          d = √((x2 – x1)^2 (y2 – y1)^2)

where ‘d’ is the distance between the points (x1, y1) and (x2, y2). This formula can be used to find the distance between any two points in a Cartesian coordinate system.

It is important to note that the distance formula is derived from the Pythagorean theorem.

Comprehensive Guide to Understanding and Using the Definition, Formula, Examples, and Applications in Geometry and Calculus

1. Section Formula:

The section formula is used to find the coordinates of a point that divides a given line segment in a given ratio. The formula is given by

Let A (x1, y1) and B (x2, y2) be two distinct points in a cartesian coordinate system, and let the point P (x, y) divide the line segment AB in the ratio m:n. Then the coordinates of P can be found using the following formulas:            

               x = (mx2 nx1)/(m n)

               y = (my2 ny1)/(m n)

Examples and Applications

1. To find the midpoint of a line segment: The midpoint of a line segment is the point that divides the line segment in the ratio of 1:1. So, if the coordinates of the two points of the line segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint are given by:

                     x = (x1 x2)/2 and y = (y1 y2)/2

1. To find the point of intersection of two lines, The point of intersection of two lines can be found by equating the coordinates of the point obtained from the section formula with the coordinates of the two lines.
2. To find the coordinates of a point that divides a line segment joining two given points in a given ratio.
3. To find the third vertex of a triangle if the other two vertices and the ratio in which the line joining them is divided at the third vertex are given.
4. If the equation of the line and the equation of the circle are known, it is possible to find the point of intersection of a line and a circle.

The Slope of a Line

• Definition and Formula
• Types of Lines (positive, negative, zero, and undefined slope)
• Examples and Applications

The Slope of a Line

The slope of a line, also known as the gradient of a line, is a measure of how steep a line is. It is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line. The slope of a line is represented by the letter “m” and is given by the formula:

m = (y2 – y1) / (x2 – x1).

where (x1, y1) and (x2, y2) are two distinct points on the line.

Types of Lines

1. Positive Slope: A line has a positive slope if it goes up from left to right. This means that as the x-coordinate increases, the y-coordinate also increases.
2. Negative Slope: A line has a negative slope if it goes down from left to right. This means that as the x-coordinate increases, the y-coordinate decreases.
3. Zero Slope: A line has a zero slope if it is horizontal. This means that the y-coordinate does not change as the x-coordinate increases.
4. Undefined Slope: A line has an undefined slope if it is vertical. This means that the x-coordinate does not change as the y-coordinate increases.

Examples and Applications

• Finding the slope of a line given two points The slope of a line can be found using the formula m = (y2 – y1) / (x2 – x1) by substituting the coordinates of the two points into the formula.
• Finding the equation of a line The slope-intercept form of the equation of a line is y = mx b, where m is the slope of the line and b is the y-intercept.
• Determining the nature of lines: By finding the slope of a line, we can decide whether or not it has a positive slope, a negative slope, a zero slope, or an undefined slope.
• To find the angle between two lines.
• To find the angle between a line and the x-axis.

Midpoint Formula

• Definition and Formula
• Examples and Applications

Midpoint Formula:

The centre of a cord segment is the end that is precisely halfway between the two endpoints of the cord segment. The midpoint formula is a mathematical expression used to find the coordinates of the midpoint of a line segment. The formula is given by

x = (x1 x2)/2

y = (y1 y2)/2

Where (x1, y1) and (x2, y2) are the coordinates of the two endpoints of the line segment.

Examples and Applications:

Finding the Midpoint of a Line Segment: The coordinates of the midpoint of a line segment can be found by substituting the coordinates of the two endpoints of the line segment into the midpoint formula.

• To find the equation of the line passing through the midpoint of a line segment joining two given points.
• To find the equation of the line parallel to a given line and passing through the midpoint of a line segment joining two given points.
• To find the equation of the line perpendicular to a given line and passing through the midpoint of a line segment joining two given points.
• To find the equation of the bisector of the angle between two lines passing through the midpoint of a line segment joining two given points.

Equation of a Line

• Slope-intercept Form
• Point-slope Form
• Examples and Applications

Equation of a Line

An equation of a line is a mathematical expression that describes the relationship between the x and y coordinates of a line. There are two common forms of the equation of a line: the slope-intercept form and the point-slope form.

• Slope-Intercept Form: The slope-intercept form of the equation of a line is y = mx b, where m is the slope of the line and b is the y-intercept. The y-intercept is the point where the line crosses the y-axis and is represented by the term b. This form of the equation is useful when the slope of the line and the y-intercept are known.

• Point-Slope Form: The point-slope form of the equation of a line is y y1 = m(x x1), where m is the slope of the line and (x1, y1) is a point on the line. This form of the equation is useful when the slope of the line and the coordinates of one point on the line are known.

Examples and Applications:

• Finding the equation of a line given the slope and y-intercept The slope-intercept form of the equation of a line can be used to find the equation of a line when the slope and y-intercept are known.
• Finding the equation of a line given the slope and a point on the line The point-slope form of the equation of a line can be used to find the equation of a line when the slope and the coordinates of one point on the line are known.
• To find the equation of the line parallel to a given line and passing through a given point.
• To find the equation of the line perpendicular to a given line and passing through a given point.
• To find the equation of the perpendicular bisector of a line segment joining two given points.

Distance of a Point from a Line

• Definition and Formula
• Examples and Applications

Distance of a Point from a Line

The distance of a point from a line is the shortest distance between a point and a line in a two-dimensional space. The formula for the distance of a point P(x1,y1) from a line represented by the equation axe by c = 0 is given by:

d = |ax1 by1 c| / (a2 b2)

Where a, b, and c are the coefficients of the equation of the line, and (x1, y1) are the coordinates of the point P.

Examples and Applications:

• To find the distance of a point from a line given the equation of the line and the coordinates of the point, By substituting the coefficients of the equation of the line and the coordinates of the point into the distance formula, we can find the distance between the point and the line.
• To find the equation of the line passing through a point and at a given distance from another point.
• To find the equation of the line passing through two points and at a given distance from a third point.
• To find the equation of the line passing through a point and being parallel to a given line.
• To find the equation of the line passing through a point and being perpendicular to a given line.

The Straight Line

• Definitions and Properties
• Distance Between Two Parallel Lines
• Distance Between a Point and a Line
• Examples and Applications

The Straight Line

A straight line is a geometric object that is defined by its direction and location in two-dimensional space. It is characterised by having no curvature, and all its points are equally distant from each other.

Definition and Properties:

• A straight line can be described as the shortest length between two points.
• A straight line is infinitely long and extends in both directions.
• A straight line can be represented by the equation y = mx b, where m is the slope and b is the y-intercept.
• Two distinct lines are parallel if and only if they have the same slope.
• Two distinct lines are perpendicular if and only if the product of their slopes is -1.
• The distance between two parallel lines is the shortest distance between the two lines. If the equation of the two lines is given by y = mx b and y = mx c, where m is the same for both lines, then the distance between the lines is |b-c| / (m2 1).

Distance Between a Point and a Line

The distance between a point and a line is the shortest distance between the point and the line. If a point P(x1, y1) and the equation of a line are y = mx b, then the distance between the point P and the line is |m*x1 – y1 b| / (m2 1).

Examples and Applications:

• To find the equation of a line passing through two given points, use the slope-intercept form of the equation of a line, y=mx b, and by using the coordinates of the two points, we can find the slope (m) and the y-intercept (b).
• To find the distance between two parallel lines: by finding the slope of the two lines and if they are equal, then the distance between them is calculated by the formula provided above.
• To find the distance between a point and a line: by substituting the point’s coordinates and the equation of the line into the distance formula provided above.
• To find the equation of the line parallel to a given line and passing through a given point, find the slope of the given line and use it in the point-slope form of the equation of a line.
• To find the equation of the line perpendicular to a given line and passing through a given point, find the negative reciprocal of the slope of the given line and use it in the point-slope form of the equation of a line.

Key Takeaways from the Chapter

1. The Cartesian coordinate system is a two-dimensional system that is used to plot points and graph lines and shapes.
2. The distance formula is used to find the distance between two points in the coordinate plane.
3. The section formula is used to find the coordinates of a point that divides a line segment into a specific ratio.
4. The slope of a line is a measure of how steep a line is and can be positive, negative, zero, or undefined.
5. The midpoint formula is used to find the coordinates of the midpoint of a line segment.
6. The equation of a line can be represented in two forms: the slope-intercept form and the point-slope form.
7. A straight line is a geometric object that is defined by its direction and location in a two-dimensional space.

Common Mistakes to Avoid

1. Not understanding the difference between slope and distance
2. You need to learn how to use the distance and section formulas correctly.
3. Not understanding the concept of slope and how to calculate it.
4. Not understanding the concept of parallel and perpendicular lines.
5. Not understanding the difference between the slope-intercept form and the point-slope form of the equation of a line.
6. Need help understanding the concept of the distance of a point from a line. Practice problems and solutions.