Problem 55 - Calculus Single Variable

Question

Find a point \((a, b)\) so that the line through \((a, b)\) and (-2,7) is perpendicular to the line through (-2,7) and (4,9).

Step-by-Step Solution

Answer

One point is \((0, 1)\).

Step 1: Find the Slope of the Given Line

The line through (-2,7) and (4,9) requires calculating the slope. The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). For points (-2,7) and (4,9), calculate the slope: \[m_1 = \frac{9 - 7}{4 + 2} = \frac{2}{6} = \frac{1}{3}.\] Thus, the slope of the first line \(m_1\) is \(\frac{1}{3}.\)

Step 2: Determine the Slope of the Perpendicular Line

Two lines are perpendicular if the product of their slopes \(m_1\) and \(m_2\) is \(-1\). Therefore, for the line perpendicular to the first line, calculate \(m_2\) as: \[m_1 \times m_2 = -1.\] Substitute \(m_1 = \frac{1}{3}\) into the equation: \[\frac{1}{3} \times m_2 = -1.\] Solve for \(m_2\): \[m_2 = -3.\] The slope of the perpendicular line \(m_2\) is \(-3\).

Step 3: Use Point-Slope Form to Find Equation of the Perpendicular Line

Use the point-slope form of a line, \(y - y_1 = m(x - x_1)\), with point (-2,7) and slope \(-3\), to find the equation of the line: \[y - 7 = -3(x + 2).\] Expand and simplify the equation: \[y - 7 = -3x - 6,\] leading to \[y = -3x + 1.\] This is the equation of the line through (-2,7) perpendicular to the original line.

Step 4: Choose Any Point on the Perpendicular Line

Any point \((a, b)\) on the line \(y = -3x + 1\) satisfies the condition. Select any convenient value for \(x\) and solve for \(y\). For instance, let \(x = 0\), then \(y = -3(0) + 1 = 1\). Thus, one such point can be \((0, 1)\).

Key Concepts

Understanding Slope Calculation

Calculating the slope of a line is a foundational skill in geometry. The slope is a measure of how steep a line is. To find the slope between two points on a line, you use the formula:

• Identify the coordinates of the two points:
  • Let’s call the first point \(x_1, y_1\), and the second point \(x_2, y_2\).
• Apply the slope formula:
  • \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

In our exercise, we calculated the slope between points (-2,7) and (4,9).

• By plugging in the values, we got \(m_1 = \frac{9 - 7}{4 + 2} = \frac{2}{6} = \frac{1}{3}\).

This slope tells you that for every increase of 3 units along the x-axis, the y-axis increases by 1 unit.

Exploring Point-Slope Form

The point-slope form is a useful way to express the equation of a line if you know a point on the line and its slope. It is written as: \(y - y_1 = m(x - x_1)\), where:

• \(y_1\) and \(x_1\) are the coordinates of a given point on the line.
• \(m\) is the slope of the line.

To determine the equation of our perpendicular line in the exercise, we used the point (-2,7) and the slope \(-3\).

• Substitute these values into the point-slope form:
• \(y - 7 = -3(x + 2)\).

This setup allowed us to transform the basic line equation into a more usable form by solving for \(y\).

Crafting the Line Equation

Once the basic form of a line is known, the next step is to simplify it to understand the full equation of the line.

• From the point-slope form \(y - 7 = -3(x + 2)\), you expand it. Simply distribute the multiplier:
• \(y - 7 = -3x - 6\)
• Simplify further to achieve the slope-intercept form of the line:
• \(y = -3x + 1\)

This final form clearly shows both the slope and the y-intercept of the line, making it easier to interpret its behavior on a graph, where it will cross the y-axis at point (0,1).
This simple format is particularly helpful for graphing and provides a quick view of how the line moves across the coordinate plane.

Geometry of Perpendicular Lines

In geometry, understanding perpendicular lines is crucial. Two lines are perpendicular if they intersect to form a right angle (90 degrees). A unique feature of perpendicular lines is their slopes. The slopes of two perpendicular lines are negative reciprocals of each other. If one line’s slope is \(\frac{1}{3}\), the other’s will be \(-3\) to maintain the relationship that their product equals \(-1\).

• \(m_1 \times m_2 = -1\).

In this exercise, this concept helped us identify the slope of our new line and assure its perpendicularity to the initial one.
Such geometric understanding strengthens the reasoning behind slope calculations, turning abstract numbers into tangible angles and relationships between lines.