Unit 7: Coordinate Geometry

Coordinate geometry provides a powerful connection between algebra and geometry through the use of graphs and curves on a Cartesian plane. It allows for geometric problems, such as finding distances and areas, to be solved using algebraic equations. By utilizing a system of perpendicular axes, any position in a plane can be represented as a unique pair of numerical coordinates.

7.1 Distance Between Two Points

• A coordinate system (or Cartesian plane) is divided into four quadrants by the x-axis (horizontal line) and the y-axis (vertical line).
• The point where these axes intersect is the origin, denoted by the ordered pair (0, 0).
• Distance is always a positive value, or zero if the two points coincide.
• For points aligned horizontally, the distance is the absolute difference of their x-coordinates: $d = |x_{2} – x_{1}|$.
• For points aligned vertically, the distance is the absolute difference of their y-coordinates: $d = |y_{2} – y_{1}|$.
• The general distance formula for any two points $P(x_{1}, y_{1})$ and $Q(x_{2}, y_{2})$ is derived from the Pythagoras Theorem: $d = \sqrt{(x_{2} – x_{1})^{2} + (y_{2} – y_{1})^{2}}$.

7.2 Division of a Line Segment

• A line segment is a portion of a line with two distinct endpoints.
• Midpoint: The point R(x, y) that divides a line segment into two equal parts.
• Midpoint Formula: $R(x, y) = \left(\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}\right)$.
• Section Formula: A point $R(x_{0}, y_{0})$ that divides a line segment internally in the ratio $p : q$ is found using:
  • $x_{0} = \frac{px_{2} + qx_{1}}{p + q}$
  • $y_{0} = \frac{py_{2} + qy_{1}}{p + q}$

7.3 Equation of a Line

• Gradient (slope): A number describing the steepness of a line, defined as the ratio of the “vertical rise” to the “horizontal run”.
• The slope (m) of a non-vertical line through points $P(x_{1}, y_{1})$ and $Q(x_{2}, y_{2})$ is: $m = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}$.
• A horizontal line has a slope of 0.
• A vertical line has no defined slope.
• Angle of inclination ($\theta$): The angle measured anticlockwise from the positive x-axis to the line, where $0^{\circ} \le \theta < 180^{\circ}$.
• Relationship between slope and inclination: $m = \tan(\theta)$.

Forms of the Equation of a Line:

• Point-slope form: $y – y_{1} = m(x – x_{1})$.
• Slope-intercept form: $y = mx + b$, where b is the y-intercept.
• Two-point form: $y – y_{1} = \frac{y_{2} – y_{1}}{x_{2} – x_{1}}(x – x_{1})$.
• General equation of a line: $Ax + By + C = 0$.

7.4 Parallel and Perpendicular Lines

• Parallel lines: Two non-vertical lines are parallel if and only if they have the same slope ($m_{1} = m_{2}$).
• Perpendicular lines: Two non-vertical lines are perpendicular if and only if the product of their slopes is -1 ($m_{1} \times m_{2} = -1$).

7.5 Equation of a Circle

• A circle is the locus of points equidistant from a fixed center.
• Standard Form: The equation of a circle with center (h, k) and radius r is: $(x – h)^{2} + (y – k)^{2} = r^{2}$.
• If the center is at the origin (0, 0), the equation simplifies to: $x^{2} + y^{2} = r^{2}$.

Process: Converting General Circle Equations to Standard Form

1. Group the x-terms together and the y-terms together.
2. Move the constant term to the right side of the equation.
3. Use the completing the square method for both the x and y groupings.
4. Add the necessary constants to both sides of the equation to maintain balance.
5. Factor the resulting trinomials into squared binomials to identify the center (h, k) and the radius squared ($r^{2}$).

7.6 Applications

• Area of a Triangle: The area of a triangle with vertices $(x_{1}, y_{1})$,$(x_{2}, y_{2})$ , and $(x_{3}, y_{3})$ is: $A = \frac{1}{2} |x_{1}(y_{2} – y_{3}) + x_{2}(y_{3} – y_{1}) + x_{3}(y_{1} – y_{2})|$.
• Coordinate geometry is used to prove geometric properties, such as showing points are collinear or identifying types of triangles (isosceles, equilateral, right-angled) based on side lengths.

Key Terminology

• Angle of inclination: The angle formed between the positive x-axis and a line in the anticlockwise direction.
• Center: The fixed point from which all points on a circle are equidistant.
• Collinear: Points that lie on the same straight line.
• Coordinate geometry: The study of geometry using a coordinate system and algebraic principles.
• Coordinates: The ordered pair (x, y) that describes a point’s position on a plane.
• Equation of a line: An algebraic representation of a straight line on a coordinate plane.
• General equation of a line: The form $Ax + By + C = 0$ representing a line.
• Gradient: The measure of the steepness of a line; also known as the slope.
• Horizontal line: A line parallel to the x-axis with a slope of zero.
• Mid-point: The point exactly in the middle of a line segment.
• Non-vertical line: A line that is not parallel to the y-axis and has a definable slope.
• Point-slope form: A line equation form using a point $(x_{1}, y_{1})$ and a slope m.
• Radius: The distance from the center of a circle to any point on its boundary.
• Slope: The ratio of the vertical change to the horizontal change between two points.
• Slope-intercept form: A line equation form $y = mx + b$ showing slope and y-intercept.
• Steepness: The degree to which a line or hill rises or falls.
• Two-point form: A line equation form determined using two distinct points.
• Vertical line: A line parallel to the y-axis with no defined slope.