Introduction

Coordinate geometry is a branch of mathematics that deals with the study of geometric figures using coordinates. It involves representing points on a plane using ordered pairs of numbers, usually denoted as $(x, y)$, where $x$ represents the horizontal position (abscissa) and $y$ represents the vertical position (ordinate). By using coordinates, we can determine properties of geometric shapes, such as distances between points, slopes of lines, and equations of curves.

Distance Formula

The distance between two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in a coordinate plane can be calculated using the distance formula: $$ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

Example: Find the distance between points $A(3, 5)$ and $B(7, 9)$. $$ \text{Distance} = \sqrt{(7 - 3)^2 + (9 - 5)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} $$

Slope of a Line

The slope of a line passing through two points $A(x_1, y_1)$ and $B(x_2, y_2)$ is given by: $$ \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1} $$

Example: Find the slope of the line passing through points $A(2, 4)$ and $B(6, 10)$. $$ \text{Slope} = \frac{10 - 4}{6 - 2} = \frac{6}{4} = \frac{3}{2} $$

Midpoint Formula

The coordinates of the midpoint of a line segment with endpoints $A(x_1, y_1)$ and $B(x_2, y_2)$ can be found using the midpoint formula: $$ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) $$

Example: Find the midpoint of the line segment with endpoints $A(-1, 3)$ and $B(5, -1)$. $$ \text{Midpoint} = \left(\frac{-1 + 5}{2}, \frac{3 + (-1)}{2}\right) = (2, 1) $$

Equation of a Line

The equation of a line in the form $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept. If we know a point $(x_1, y_1)$ on the line, we can find the equation using: $$ y - y_1 = m(x - x_1) $$

Example: Find the equation of the line passing through point $(-2, 4)$ with slope $\frac{1}{3}$. $$ y - 4 = \frac{1}{3}(x + 2) \Rightarrow y = \frac{1}{3}x + \frac{10}{3} $$

Circle Equation

The equation of a circle with center $(h, k)$ and radius $r$ is given by: $$ (x - h)^2 + (y - k)^2 = r^2 $$

Example: Find the equation of a circle with center at $(2, -3)$ and radius 5. $$ (x - 2)^2 + (y + 3)^2 = 25 $$

Common Mistakes

• Forgetting to square the differences in the distance formula.
• Incorrectly calculating the slope by swapping the $x$ and $y$ coordinates.
• Misinterpreting the midpoint formula and not dividing by 2.

Key Points

• Coordinate geometry uses ordered pairs of numbers to represent points on a plane.
• The distance formula calculates the distance between two points.
• The slope formula determines the inclination of a line.
• The midpoint formula finds the middle point of a line segment.
• Equations of lines and circles can be derived using specific formulas.

Practice Questions

1. Find the distance between points $P(1, -2)$ and $Q(4, 5)$.
2. Determine the slope of the line passing through points $C(-3, 2)$ and $D(1, 8)$.
3. Calculate the midpoint of the line segment with endpoints $E(0, -4)$ and $F(6, 2)$.
4. Write the equation of the line with slope $\frac{2}{3}$ passing through point $G(4, -1)$.
5. Find the equation of a circle with center at $(3, -1)$ and radius 4.

Practice Questions - Answers

1. The distance between $P(1, -2)$ and $Q(4, 5)$ is: $$ \sqrt{(4 - 1)^2 + (5 - (-2))^2} = \sqrt{9 + 49} = \sqrt{58} $$
2. The slope of the line passing through $C(-3, 2)$ and $D(1, 8)$ is: $$ \frac{8 - 2}{1 - (-3)} = \frac{6}{4} = \frac{3}{2} $$
3. The midpoint of $E(0, -4)$ and $F(6, 2)$ is: $$ \left(\frac{0 + 6}{2}, \frac{-4 + 2}{2}\right) = (3, -1) $$
4. The equation of the line passing through $G(4, -1)$ with slope $\frac{2}{3}$ is: $$ y - (-1) = \frac{2}{3}(x - 4) \Rightarrow y = \frac{2}{3}x + \frac{5}{3} $$
5. The equation of the circle with center at $(3, -1)$ and radius 4 is: $$ (x - 3)^2 + (y + 1)^2 = 16 $$

Introduction

Coordinate geometry is a branch of mathematics that deals with the study of geometry using algebraic methods. In this topic, we use the Cartesian coordinate system to represent points, lines, and shapes on a plane. The Cartesian coordinate system consists of two perpendicular axes, the x-axis, and the y-axis, which intersect at the origin (0, 0).

Coordinates and Points

• Coordinates: Coordinates are pairs of numbers that represent the location of a point in a plane.
• Point: A point is represented by a pair of coordinates (x, y), where x is the distance along the x-axis and y is the distance along the y-axis.

Example: Find the coordinates of point A in the Cartesian plane if A is located at (3, 4). Solution: The coordinates of point A are (3, 4).

Distance Formula

• Distance Formula: The distance between two points A(x1, y1) and B(x2, y2) in a plane is given by: $$ \sqrt{(x2 - x1)^2 + (y2 - y1)^2} $$

Example: Find the distance between points A(2, 3) and B(5, 7). Solution: Using the distance formula: $$ \sqrt{(5 - 2)^2 + (7 - 3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$

Midpoint Formula

• Midpoint Formula: The midpoint of a line segment with endpoints A(x1, y1) and B(x2, y2) is given by: $$ \left(\frac{x1 + x2}{2}, \frac{y1 + y2}{2}\right) $$

Example: Find the midpoint of the line segment with endpoints A(1, 3) and B(7, 9). Solution: Using the midpoint formula: $$ \left(\frac{1 + 7}{2}, \frac{3 + 9}{2}\right) = (4, 6) $$

Slope of a Line

• Slope: The slope of a line passing through two points A(x1, y1) and B(x2, y2) is given by: $$ \frac{y2 - y1}{x2 - x1} $$

Example: Find the slope of the line passing through points A(2, 3) and B(4, 7). Solution: Using the slope formula: $$ \frac{7 - 3}{4 - 2} = \frac{4}{2} = 2 $$

Equation of a Line

• Equation of a Line: The equation of a line passing through a point (x1, y1) with slope m is given by the point-slope form: $$ y - y1 = m(x - x1) $$

Example: Find the equation of the line passing through point (2, 3) with slope 4. Solution: Using the point-slope form: $$ y - 3 = 4(x - 2) $$ $$ y - 3 = 4x - 8 $$ $$ y = 4x - 5 $$

Common Mistakes

• Forgetting to square the differences when calculating distances.
• Misinterpreting the signs when finding the slope of a line.
• Incorrectly applying the midpoint formula by forgetting to divide by 2.

Key Points

• Coordinates are pairs of numbers representing the location of a point.
• The distance between two points is calculated using the distance formula.
• The midpoint of a line segment is found using the midpoint formula.
• The slope of a line is determined by the difference in y-coordinates over the difference in x-coordinates.
• The equation of a line can be found using the point-slope form.

Practice Questions

1. Find the distance between points A(1, 2) and B(5, 8).

Solution: Using the distance formula: $$ \sqrt{(5 - 1)^2 + (8 - 2)^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 2\sqrt{13} $$

2. Find the midpoint of the line segment with endpoints C(2, 4) and D(6, 10).

Solution: Using the midpoint formula: $$ \left(\frac{2 + 6}{2}, \frac{4 + 10}{2}\right) = (4, 7) $$

3. Determine the slope of the line passing through points E(3, 5) and F(7, 9).

Solution: Using the slope formula: $$ \frac{9 - 5}{7 - 3} = \frac{4}{4} = 1 $$

4. Find the equation of the line passing through point G(4, 6) with slope -2.

Solution: Using the point-slope form: $$ y - 6 = -2(x - 4) $$ $$ y - 6 = -2x + 8 $$ $$ y = -2x + 14 $$

5. Given points H(0, 2) and I(3, 6), find the distance between them.

Solution: Using the distance formula: $$ \sqrt{(3 - 0)^2 + (6 - 2)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $$