AOS1 Topic 1: Coordinate Geometry (Part 2)

Coordinate geometry (or analytic geometry) is defined as the study of geometry using the coordinate points. Using coordinate geometry, it is possible to find the distance between two points, dividing lines in m:n ratio, finding the mid-point of a line, calculating the area of a triangle in the Cartesian plane, etc.

Coordinate Geometry Terms

Cartesian Coordinate System:

Consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). The point where the axes intersect is called the origin, denoted as \((0, 0)\). Any point in the plane can be described by an ordered pair \((x, y)\), where \(x\) is the horizontal distance from the origin and \(y\) is the vertical distance from the origin.

Distance between two points

The distance between two points \(A(x_1, y_1)\) and \(B(x_2, y_2)\) is given by:

\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

Midpoint of a line segment

The midpoint of the line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is the point with coordinates:

\[ \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Gradient of a straight line

The gradient \(m\) of a straight line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Equation of a straight line

Gradient-intercept form: A straight line with gradient \(m\) and y-axis intercept \(c\) has the equation:

\[ y = mx + c \]

Point-gradient form: The equation of a straight line passing through a given point \( (x_1, y_1) \) and having gradient \(m\) is:

\[ y - y_1 = m(x - x_1) \]

Two-point form: The equation of a straight line passing through two given points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[ y - y_1 = m(x - x_1) \]

where \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

Intercept form: The straight line passing through the two points \( (a, 0) \) and \( (0, b) \) has the equation:

\[ \frac{x}{a} + \frac{y}{b} = 1 \]

Tangent of the angle of slope

For a straight line with gradient \(m\), the angle of slope is found using:

\[ m = \tan \theta \]

where \( \theta \) is the angle that the line makes with the positive direction of the x-axis.

Perpendicular straight lines

If two straight lines are perpendicular to each other, the product of their gradients is \(-1\), i.e., \( m_1 m_2 = -1 \) (unless one line is vertical and the other horizontal).

Coordinate geometry is a powerful tool for solving geometric problems and proving geometric theorems using algebraic methods. It bridges the gap between algebra and geometry, providing a deeper understanding of both subjects.

What is a Co-ordinate and a Co-ordinate Plane?

You must be familiar with plotting graphs on a plane, from the tables of numbers for both linear and non-linear equations. The number line which is also known as a Cartesian plane is divided into four quadrants by two axes perpendicular to each other, labelled as the x-axis (horizontal line) and the y-axis (vertical line).

The four quadrants along with their respective values are represented in the graph below:

Quadrant 1: (+x, +y)
Quadrant 2: (-x, +y)
Quadrant 3: (-x, -y)
Quadrant 4: (+x, -y)

The point at which the axes intersect is known as the origin. The location of any point on a plane is expressed by a pair of values (x, y) and these pairs are known as the coordinates.

Coordinate Geometry Theorems

Midpoint Theorem: To Find Mid-point of a Line Connecting Two Points

Consider the same points A and B, which have coordinates \((x_1, y_1)\) and \((x_2, y_2)\), respectively. Let M(x,y) be the midpoint of lying on the line connecting these two points A and B. The coordinates of point M is given as:

\[ M \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

Angle Formula: To Find The Angle Between Two Lines

Consider two lines A and B, having their slopes \(m_1\) and \(m_2\), respectively. Let “θ” be the angle between these two lines, then the angle between them can be represented as:

\[ \tan(\theta) = \left| \frac{m_2 - m_1}{1 + m_1 m_2} \right| \]

Special Cases:

Case 1: When the two lines are parallel to each other,
\(m_1 = m_2 = m\)

Substituting the value in the equation above,

\[ \tan(\theta) = \left| \frac{m - m}{1 + m \cdot m} \right| = 0 \]

Case 2: When the two lines are perpendicular to each other,
\(m_1 \cdot m_2 = -1\)

Substituting the value in the original equation,

\[ \tan(\theta) = \left| \frac{-1 - m_1 m_2}{1 + m_1 m_2} \right| = \text{undefined} \]

\( \Rightarrow \theta = 90^\circ \)

Section Formula: To Find a Point Which Divides a Line into m:n Ratio

Consider a line A and B having coordinates \((x_1, y_1)\) and \((x_2, y_2)\), respectively. Let P be a point that divides the line in the ratio m:n, then the coordinates of the point P is given as:

When the ratio m:n is internal:

\[ P \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right) \]

When the ratio m:n is external:

\[ P \left( \frac{mx_2 - nx_1}{m-n}, \frac{my_2 - ny_1}{m-n} \right) \]

Students can follow the link provided to learn more about the section formula along its proof and solved examples.

Area of a Triangle in Cartesian Plane

The area of a triangle in coordinate geometry whose vertices are \((x_1, y_1)\), \((x_2, y_2)\) and \((x_3, y_3)\) is:

\[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]

If the area of a triangle whose vertices are \((x_1, y_1)\), \((x_2, y_2)\) and \((x_3, y_3)\) is zero, then the three points are collinear.

Example 1

A straight line passes through the points A(−2, 6) and B(4, 7). Find:

  1. the distance AB
  2. the midpoint of line segment AB
  3. the gradient of line AB
  4. the equation of line AB
  5. the equation of the line parallel to AB which passes through the point (1, 5)
  6. the equation of the line perpendicular to AB which passes through the midpoint of AB

Solution:

Given Points:

A(−2, 6)

B(4, 7)

(a) The Distance AB

The distance between two points (x1, y1) and (x2, y2) is given by:

d = √((x2 - x1)2 + (y2 - y1)2)

Substitute A(−2, 6) and B(4, 7):

d = √((4 - (−2))2 + (7 - 6)2)

d = √((4 + 2)2 + (1)2)

d = √(62 + 12)

d = √(36 + 1)

d = √37

So, the distance AB is √37.

(b) The Midpoint of Line Segment AB

The midpoint M of a line segment joining points (x1, y1) and (x2, y2) is given by:

M = ((x1 + x2)/2, (y1 + y2)/2)

Substitute A(−2, 6) and B(4, 7):

M = ((−2 + 4)/2, (6 + 7)/2)

M = (2/2, 13/2)

M = (1, 13/2)

So, the midpoint of AB is (1, 13/2).

(c) The Gradient of Line AB

The gradient (slope) m of a line passing through points (x1, y1) and (x2, y2) is given by:

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

Substitute A(−2, 6) and B(4, 7):

m = (7 - 6) / (4 - (−2))

m = 1 / (4 + 2)

m = 1 / 6

So, the gradient of line AB is 1/6.

(d) The Equation of Line AB

We know the points A(−2, 6) and B(4, 7). To find the equation of the line passing through these points, we first find the gradient (slope) m:

m = (y2 - y1) / (x2 - x1) = (7 - 6) / (4 - (−2)) = 1 / 6

Using the point-slope form of the line equation y − y1 = m(x − x1) with point A(−2, 6):

y − 6 = (1/6)(x − (−2))

Simplify the equation:

y − 6 = (1/6)(x + 2)

y − 6 = (1/6)x + 1/3

6(y − 6) = x + 2

6y − 36 = x + 2

6y − x − 38 = 0

So, the equation of line AB is 6y − x − 38 = 0.

(e) The Equation of the Line Parallel to AB Passing Through (1, 5)

Parallel lines have the same gradient, so the gradient m is 1/6. Using the point-slope form with point (1, 5):

y − 5 = (1/6)(x − 1)

Simplify the equation:

y − 5 = (1/6)(x − 1)

y − 5 = (1/6)x − 1/6

6(y − 5) = x − 1

6y − 30 = x − 1

6y − x − 29 = 0

So, the equation of the line parallel to AB passing through (1, 5) is 6y − x − 29 = 0.

(f) The Equation of the Line Perpendicular to AB Passing Through the Midpoint of AB

The midpoint of AB is (1, 13/2). The gradient of a line perpendicular to AB is the negative reciprocal of 1/6, which is -6.

Using the point-slope form with point (1, 13/2):

y − 13/2 = -6(x − 1)

Simplify the equation:

y − 13/2 = -6(x − 1)

y − 13/2 = -6x + 6

2(y − 13/2) = 2(-6x + 6)

2y − 13 = -12x + 12

2y + 12x − 25 = 0

So, the equation of the line perpendicular to AB passing through the midpoint is 2y + 12x − 25 = 0.

Example 2

Hydroponic Tomatoes Cost Calculation

A fruit and vegetable wholesaler sells 30 kg of hydroponic tomatoes for $148.50 and sells 55 kg of hydroponic tomatoes for $247.50. Find a linear model for the cost, $C, of x kg of hydroponic tomatoes. How much would 20 kg of tomatoes cost?

Solution

Let \((x_1, C_1) = (30, 148.5)\) and \((x_2, C_2) = (55, 247.5)\).

The equation of the straight line is given by:

\(C - C_1 = m(x - x_1)\) where \(m = \frac{C_2 - C_1}{x_2 - x_1}\)

Now \(m = \frac{247.5 - 148.5}{55 - 30} = 3.96\)

Therefore, \(C - 148.5 = 3.96(x - 30)\)

So, the straight line has the equation: \(C = 3.96x + 29.7\)

Substitute \(x = 20\):

\(C = 3.96 \times 20 + 29.7 = 108.9\)

Hence, it would cost $108.90 to buy 20 kg of tomatoes.

Example 3

Midpoint Calculation

Find the coordinates of \( M \), the midpoint of \( AB \), where \( A \) and \( B \) have the following coordinates: A(1, 4), B(5, 11).

Solution:

To find the coordinates of the midpoint \( M \) of line segment \( AB \) where \( A \) and \( B \) have given coordinates, you can use the midpoint formula:

\[ M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right) \]

For the coordinates \( A(1, 4) \) and \( B(5, 11) \):

Let \( A(1, 4) \) and \( B(5, 11) \).

Apply the midpoint formula:

\[ M = \left( \frac{1 + 5}{2}, \frac{4 + 11}{2} \right) \]

Calculate the coordinates:

\[ M = \left( \frac{6}{2}, \frac{15}{2} \right) \]

\[ M = (3, 7.5) \]

So, the coordinates of the midpoint \( M \) are \( (3, 7.5) \).

Example 4

Equation of a Straight Line

Find the equations of the following straight lines:

Gradient \( m = +2 \), passing through the Point \( (4, 2) \)

Solution:

To find the equation of the straight line, we use the point-gradient form of the equation of a straight line:

\[ y - y_1 = m(x - x_1) \]

Substitute the given values into the point-gradient form:

\[ y - 2 = 2(x - 4) \]

Now, simplify the equation:

\[ y - 2 = 2x - 8 \]

\[ y = 2x - 8 + 2 \]

\[ y = 2x - 6 \]

Therefore, the equation of the straight line with gradient +2 passing through the point (4, 2) is:

\[ y = 2x - 6 \]

Example 5

Find the equation of the line passing through the point (−1, 3) which is:

Parallel to the line with equation 2x + 5y − 10 = 0

Solution:

  1. Find the gradient of the given line:

    Rewrite the equation \(2x + 5y - 10 = 0\) in slope-intercept form:

    \(5y = -2x + 10\)

    \(y = -\frac{2}{5}x + 2\)

    The gradient \(m\) is \(-\frac{2}{5}\).

  2. Use the point-gradient form of the equation of a line:

    Point-gradient form: \(y - y_1 = m(x - x_1)\)

    Given point \((x_1, y_1) = (-1, 3)\)

    Substitute the values:

    \(y - 3 = -\frac{2}{5}(x - (-1))\)

    \(y - 3 = -\frac{2}{5}(x + 1)\)

    \(y - 3 = -\frac{2}{5}x - \frac{2}{5}\)

    \(y = -\frac{2}{5}x - \frac{2}{5} + 3\)

    \(y = -\frac{2}{5}x - \frac{2}{5} + \frac{15}{5}\)

    \(y = -\frac{2}{5}x + \frac{13}{5}\)

    So, the equation is \(y = -\frac{2}{5}x + \frac{13}{5}\).


Example 6

Find the equation of the line which passes through the point of intersection of the lines y = x and x + y = 6 and which is perpendicular to the line with equation 3x + 6y = 12.

Solution:

  1. Find the point of intersection:

    Solve the equations \(y = x\) and \(x + y = 6\) simultaneously:

    Substitute \(y\) with \(x\) in \(x + y = 6\):

    \(x + x = 6\)

    \(2x = 6\)

    \(x = 3\)

    Substitute \(x = 3\) back into \(y = x\):

    \(y = 3\)

    So, the point of intersection is \((3, 3)\).

  2. Find the gradient of the perpendicular line:

    The equation of the given line is \(3x + 6y = 12\). Rewrite this in slope-intercept form:

    \(6y = -3x + 12\)

    \(y = -\frac{1}{2}x + 2\)

    The gradient \(m\) of this line is \(-\frac{1}{2}\).

    The gradient of the line perpendicular to this will be the negative reciprocal of \(-\frac{1}{2}\), which is \(2\).


  • Find the equation of the line:

    The gradient \(m\) of the line we are looking for is \(2\). We also know that this line passes through the point \((3, 3)\).

    Using the point-gradient form \(y - y_1 = m(x - x_1)\):

    \(y - 3 = 2(x - 3)\)

    Simplify this equation:

    \(y - 3 = 2x - 6\)

    \(y = 2x - 6 + 3\)

    \(y = 2x - 3\)

  • So, the equation of the line is \(y = 2x - 3\).

    Exercise &&1&& (&&1&& Question)

    If M is the midpoint of XY, find the coordinates of Y when X and M have the following coordinates:

    X(-4, 5)

    M(0, 6)

    1
    Submit

    Exercise &&2&& (&&1&& Question)

    What is the gradient in the equation: \(4x + 6y = 12\)?

    2
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    Exercise &&3&& (&&1&& Question)

    Find the distance between the following pairs of points: (2, 6), (3, 4)

    3
    Submit

    Exercise &&4&& (&&1&& Question)

    The length of the line segment joining the points A(2, -1) and B(5, y) is 5 units. Find the value of y.

    4
    Submit