Problem 19 - The Calculus with Analytic Geometry

Question

For each of the following sets of three points, determine by means of slopes if the points are on a line: (a) \((2,3)\), \((-4,-7),(5,8) ;\) (b) \((-3,6),(3,2),(9,-2) ;(\mathrm{c})(2,-1),(1,1),(3,4) ;\) and \((\mathrm{d})(4,6),(1,2),(-5,-4) .\)

Step-by-Step Solution

Answer

Sets (a) and (b) are collinear. Sets (c) and (d) are not collinear.

Step 1: Calculate Slope Between First and Second Point

Use the slope formula \ \ \(m = \frac{y_2 - y_1}{x_2 - x_1}\) \ \ For \ \ (a) \ \ \((2,3)\) and \((-4,-7)\) \ \ \[ m_1 = \frac{-7 - 3}{-4 - 2} = \frac{-10}{-6} = \frac{5}{3} \]

Step 2: Calculate Slope Between Second and Third Point

Use the slope formula \ \ \(m = \frac{y_2 - y_1}{x_2 - x_1}\) \ \ For \ \ (a) \ \ \((-4,-7)\) and \((5,8)\) \ \ \[ m_2 = \frac{8 - (-7)}{5 - (-4)} = \frac{8 + 7}{5 + 4} = \frac{15}{9} = \frac{5}{3} \]

Step 3: Compare Slopes from Step 1 and Step 2

For points to be collinear, the slopes must be equal. \ \ For \ \ (a) \ \ \[m_1 = m_2 = \frac{5}{3}\] \ \ The points (2,3), (-4,-7), (5,8) are collinear.

Step 4: Repeat Steps for Other Sets of Points

Repeat Steps 1 to 3 for sets (b), (c), and (d).\ \ For \ \ (b) \ \ \((-3,6)\) and \((3,2)\) \ \ \[ m_1 = \frac{2 - 6}{3 - (-3)} = \frac{-4}{6} = -\frac{2}{3} \] \ \ \((3,2)\) and \((9,-2)\) \ \ \[ m_2 = \frac{-2 - 2}{9 - 3} = \frac{-4}{6} = -\frac{2}{3} \] \ \ \[m_1 = m_2 = -\frac{2}{3}\] \ \ The points (-3,6), (3,2), (9,-2) are collinear.\ \ For \ \ (c) \ \ \((2,-1)\) and \((1,1)\) \ \ \[ m_1 = \frac{1 - (-1)}{1 - 2} = \frac{2}{-1} = -2 \] \ \ \((1,1)\) and \((3,4)\) \ \ \[ m_2 = \frac{4 - 1}{3 - 1} = \frac{3}{2} \] \ \ \[m_1 eq m_2\] \ \ The points (2,-1), (1,1), (3,4) are not collinear.\ \ For \ \ (d) \ \ \((4,6)\) and \((1,2)\) \ \ \[ m_1 = \frac{2 - 6}{1 - 4} = \frac{-4}{-3} = \frac{4}{3} \] \ \ \((1,2)\) and \((-5,-4)\) \ \ \[ m_2 = \frac{-4 - 2}{-5 - 1} = \frac{-6}{-6} = 1 \] \ \ \[m_1 eq m_2\] \ \ The points (4,6), (1,2), (-5,-4) are not collinear.

Key Concepts

Slope Formula

Understanding the slope formula is key when working with collinear points in analytic geometry. The slope of a line measures how steep the line is. Mathematically, the slope between any two points \((x_1, y_1)\) and \((x_2, y_2)\) on a Cartesian plane is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
This formula calculates the difference in the y-coordinates (rise) divided by the difference in the x-coordinates (run).
For example, to find the slope between the points \(2,3\) and \(-4,-7\) for part (a), we compute:
\[ m_1 = \frac{-7 - 3}{-4 - 2} = \frac{-10}{-6} = \frac{5}{3} \]
It's crucial to apply this formula correctly and carefully track each coordinate. Mastery of the slope formula helps identify patterns and understand relationships between data points.

Point Coordination

Point coordination involves working with specific values on a Cartesian plane. Each point has a unique set of coordinates expressed as \(x\) and \(y\) values in the format \((x, y)\). These values determine the exact position of the point.
When analyzing collinearity, you need at least three points. For example, examining points \((2,3)\), \((-4,-7)\), and \((5,8)\) in part (a) and checking if they lie on a straight line.
You calculate the slopes between consecutive pairs of points:
\[ m_1 = \frac{-7 - 3}{-4 - 2} = \frac{5}{3} \] \[ m_2 = \frac{8 - (-7)}{5 - (-4)} = \frac{15}{9} = \frac{5}{3} \]
Here, both slopes \(m_1\) and \(m_2\) are equal, confirming collinearity of the points.
Consistent and precise point coordination simplifies determining relationships between multiple points.

Collinear Points

Collinear points are points that lie on the same straight line. For three points to be collinear, the slopes between consecutive pairs of points must be equal.
Consider the points \(2,3\), \(-4,-7\), and \((5,8)\) in part (a). We calculate:
\[ m_1 = \frac{-7 - 3}{-4 - 2} = \frac{5}{3} \] \[ m_2 = \frac{8 - (-7)}{5 - (-4)} = \frac{15}{9} = \frac{5}{3} \]
Since \(m_1 = m_2 = \frac{5}{3}\), the points are collinear.
Applying this method to other sets of points, we see:
For set (b) \((-3,6), (3,2), (9,-2)\):
\[ m_1 = \frac{2 - 6}{3 - (-3)} = -\frac{2}{3} \]
\[ m_2 = \frac{-2 - 2}{9 - 3} = -\frac{2}{3} \]
Both slopes are equal, so the points are collinear.
Identifying non-collinear points involves checking unequal slopes:
For set (c) \((2,-1), (1,1), (3,4)\):
\[ m_1 = \frac{1 - (-1)}{1 - 2} = -2 \]
\[ m_2 = \frac{4 - 1}{3 - 1} = \frac{3}{2} \]
Since \(m_1 ≠ m_2\), the points are not collinear.
Understanding collinear points offers insight into geometric principles and relationships within graphical data.