UNIT 13 : Points, straight lines and circles in 2D
Key unit competence
Determine algebraic representations of lines, straight lines and circles in 2D.
Learning objectives
13.1 Points in 2D
Activity 13.1
What is a point?
In Junior Secondary, we studied points on a Cartesian plane. How do we define the coordinate of a point in 2D? Discuss in groups and present your findings to the rest of the class
A point is an exact position or location on a plane surface. It is important to understand that a point is not a thing, but a place. We indicate the position of a point by placing a dot with a pencil. In coordinate geometry, points are located on the plane using their coordinates - two numbers that show where the point is positioned with respect to two number line “axes” at right angles to each other.
Cartesian coordinate of a point
Distance between two points
The mid-point of a line segment
13.2 Lines in 2D
The equations of straight lines
A particular line is uniquely located in a plane if
• it has a known direction and passes through a known fixed point, or
• it passes though two known points.
Cartesian equation of a straight line
Equation of a line given a law
We can find the equation of the locus by considering a point P(x, y) on the locus and using the law to derive an equation in x and y. This will be the equation of the locus.
Parallel and perpendicular lines
If two lines are parallel, they have equal gradients
If two lines are perpendicular, the product of their gradients is –1.
Vector, parametric, scalar and Cartesian equations of the line
In 2D, a line can be defined by an equation in slope–intercept form, a vector equation, parametric equations, or a Cartesian equation (scalar equation).
Intersection of lines with equations in Cartesian form
Any point on a line has coordinates which will satisfy the equation of that line. In order to find the point in which two lines intersect we have to find a point with coordinates which satisfy both equations. This is equivalent, from an algebraic point of view, to solving the equations of the lines simultaneously.
The intersection of two lines with equations given in vector form
In order to find the point of intersection of two lines whose equations are given in vector form each equation must have a separate parameter. The method as illustrated in the following example:
13.3 Points and lines
Distance of a point from a line
Angle between two straight lines
13.4 The circle
Mental task
What is a circle? What are the main parts of a circle that you can recall? Why are they significant?
In fact the definition of a circle is: the set of all points on a plane that are at a fixed distance from a centre.
Unit circle
If we place the circle centre at (0,0) and set the radius to 1 we get:
Intersecting a line and a circle
Consider a straight line y = mx + c and a circle (x – a)2 + (y – b)2 = r2.
There are three possible situations:
1. The line cuts the circle in two distinct places, i.e. part of the line is a chord of the circle.
2. The line touches the circle, i.e. the line is a tangent to the circle.
3. The line neither cuts nor touches the circle.
Circle through three given points
Three non-collinear points define a circle,
i.e. there is one, and only one circle which can be drawn through three non-collinear points. The equation of any
