Problem 25 - The Calculus with Analytic Geometry

Question

(a) Find an equation whose graph consists of all points equidistant from the points \((-1,2)\) and \((3,4)\). (b) Draw a sketch of the graph of the equation found in (a).

Step-by-Step Solution

Answer

The equation is \y = -2x + 5\. The graph is a line passing through (1, 3) with a slope of -2.

Step 1: Identify the Concept

The problem involves finding all points equidistant from two fixed points, which forms the perpendicular bisector of the line segment joining the points \((-1,2)\) and \((3,4)\).

Step 2: Find the Midpoint

Calculate the midpoint of the line segment joining \((-1,2)\) and \((3,4)\) using the midpoint formula: \(\text{Midpoint} = \frac{ (x_1 + x_2)}{2}, \frac{(y_1 + y_2)}{2}) = \frac{-1 + 3}{2}, \frac{2 + 4}{2}) = (1,3)\).

Step 3: Calculate the Slope of the Segment

Determine the slope of the line segment joining the points: \(m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{3 - (-1)} = \frac{2}{4} = \frac{1}{2}\).

Step 4: Find the Slope of the Perpendicular Bisector

The slope of the perpendicular bisector is the negative reciprocal of the slope of the segment. Therefore, it is: \(m_{\text{perp}} = -\frac{1}{\frac{1}{2}} = -2\).

Step 5: Write the Equation of the Perpendicular Bisector

Using the point-slope form \(y - y_1 = m(x - x_1)\) and the midpoint \((1, 3)\), the equation of the perpendicular bisector is \y - 3 = -2(x - 1)\. Simplifying, \y - 3 = -2x + 2\ and \y = -2x + 5\.

Step 6: Sketch the Graph

Draw the graph of the line \y = -2x + 5\. It should pass through the midpoint \((1, 3)\), and its slope should be -2. Also, plot the original points \((-1, 2)\) and \((3, 4)\) for reference.

Key Concepts

midpoint formula

The midpoint formula is a fundamental concept in geometry. It is used to find the exact middle point of a line segment
connecting two points, \((-1,2)\) and \((3,4)\). This formula helps greatly in problems where symmetry or bisecting a line is needed.
To find the midpoint, you add the x-coordinates of both points and divide by 2 for the x-coordinate of the midpoint.
You do the same for the y-coordinates to get the y-coordinate of the midpoint.

The formula is written as:

\[ \text{Midpoint} = \frac{{ (x_1 + x_2) }}{2}, \frac{{ (y_1 + y_2) }}{2} \]

Here's how it works with our points:

• Add the x-coordinates: \(-1 + 3 = 2 \)
• Divide by 2: \(\frac{2}{2} = 1 \)
• Add the y-coordinates: \(2 + 4 = 6 \)
• Divide by 2: \(\frac{6}{2} = 3 \)

So, the midpoint is \((1, 3)\).

slope calculation

Finding the slope of a line segment between two points is key to understanding the line's inclination.
The slope tells you how steep the line is. To calculate it, you need the difference in y-coordinates divided by the difference in x-coordinates.
This is particularly useful when you need to find perpendicular lines.

The formula for the slope, \(m\), is:

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

Applying this to our points, \((-1,2)\) and \((3,4)\):

• Subtract the y-coordinates: \(4 - 2 = 2 \)
• Subtract the x-coordinates: \(3 - (-1) = 4 \)
• Divide the differences: \( \frac{2}{4} = \frac{1}{2} \)

So, the slope of the line segment is \(\frac{1}{2}\).

point-slope form

Point-slope form is one of the ways to write the equation of a line. It is especially useful when you have a point on the line
and the slope of the line. The general form of this equation is:

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

Here, \( (x_1, y_1) \) is a specific point on the line, and \(m\) is the slope.

Using the midpoint \((1, 3)\) and the perpendicular slope \(-2\):

• Plug in the values:
  \( y - 3 = -2(x - 1) \)
• Simplify:
  \( y - 3 = -2x + 2 \)
• Add 3 to both sides:
  \( y = -2x + 5 \)

The final equation of the perpendicular bisector is \( y = -2x + 5 \). This line will pass through the midpoint \((1, 3)\)
and has a slope of \(-2\).