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Chapter 9: Problem 1

Determine the distance from \(P(-4,5)\) to each of the following lines: a. \(3 x+4 y-5=0\) b. \(5 x-12 y+24=0\) c. \(9 x-40 y=0\)

Short Answer

Expert verified
The distances are 0.6 to line a, approximately 4.308 to line b, and approximately 5.756 to line c.

Step by step solution

01

Understanding the Point-to-Line Distance Formula

The distance from a point \(P(x_1, y_1)\) to a line of the form \(Ax + By + C = 0\) can be found using the formula: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] This formula calculates the perpendicular distance from the point to the line.
02

Identify Parameters for Line a

For the line \(3x + 4y - 5 = 0\), we identify \(A = 3\), \(B = 4\), and \(C = -5\). The coordinates of the point \(P\) are \((-4, 5)\).
03

Calculate Distance for Line a

Substitute the values into the distance formula: \[ d_a = \frac{|3(-4) + 4(5) - 5|}{\sqrt{3^2 + 4^2}} \] Calculate inside the absolute value: \[ 3(-4) + 4(5) - 5 = -12 + 20 - 5 = 3 \] Find the distance: \[ d_a = \frac{|3|}{\sqrt{9 + 16}} = \frac{3}{\sqrt{25}} = \frac{3}{5} = 0.6 \]
04

Identify Parameters for Line b

For the line \(5x - 12y + 24 = 0\), we identify \(A = 5\), \(B = -12\), and \(C = 24\). The coordinates of the point \(P\) are \((-4, 5)\).
05

Calculate Distance for Line b

Substitute the values into the distance formula: \[ d_b = \frac{|5(-4) - 12(5) + 24|}{\sqrt{5^2 + (-12)^2}} \] Calculate inside the absolute value: \[ 5(-4) - 12(5) + 24 = -20 - 60 + 24 = -56 \] Find the distance: \[ d_b = \frac{|-56|}{\sqrt{25 + 144}} = \frac{56}{\sqrt{169}} = \frac{56}{13} \approx 4.308 \]
06

Identify Parameters for Line c

For the line \(9x - 40y = 0\), we identify \(A = 9\), \(B = -40\), and \(C = 0\). The coordinates of the point \(P\) are \((-4, 5)\).
07

Calculate Distance for Line c

Substitute the values into the distance formula: \[ d_c = \frac{|9(-4) - 40(5) + 0|}{\sqrt{9^2 + (-40)^2}} \] Calculate inside the absolute value:\[ 9(-4) - 40(5) = -36 - 200 = -236 \] Find the distance: \[ d_c = \frac{|-236|}{\sqrt{81 + 1600}} = \frac{236}{\sqrt{1681}} = \frac{236}{41} \approx 5.756 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point-to-Line Distance Formula
The Point-to-Line Distance Formula is a powerful tool in geometry. It helps us find the shortest distance from a point to a line. This distance is always measured along the line that is perpendicular to the given line and passes through the point.
The formula to calculate this distance is: \[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \] Here,
  • \((x_1, y_1)\) are the coordinates of the point for which you're finding the distance.
  • \(A\), \(B\), and \(C\) are the coefficients from the line equation, generally presented as \(Ax + By + C = 0\).
When using this formula, you essentially compute how far the point is from the line by calculating the perpendicular drop from the point onto the line, which is the essence of the shortest distance.
Perpendicular Distance
Perpendicular distance is a key concept when talking about distances in geometry. Imagine dropping a straight line from a point to a line that is perpendicular to the line. This shortest path is the perpendicular distance.
**Why Perpendicular?**
  • It's the shortest path. Any slanted or non-perpendicular line is longer.
  • It aligns with the orthogonality of Cartesian coordinates, providing a clear, exact measure.
Analytical geometry often employs perpendicular distance to maintain accurate and precise calculations, as it is undisputed in its measurement.
Cartesian Coordinates
Cartesian Coordinates are an essential part of analytical geometry, named after Ren茅 Descartes. It relies on two perpendicular axes: the horizontal (x-axis) and the vertical (y-axis). They help in identifying the position of a point by determining how far along each axis a point is positioned.
**Key Elements:**
  • **Origin**: The point (0, 0), where the two axes intersect.
  • **Coordinates**: A pair \((x, y)\) that specifies the location with respect to the axes.
  • **Quadrants**: The two-dimensional plane is divided into four quadrants, aiding spatial understanding.
Each coordinate is expressed as a pair of numbers, which makes calculating distances, slopes, and many other geometric properties easier and more intuitive.
Analytical Geometry
Analytical Geometry, also known as coordinate geometry, is a branch of mathematics that uses algebra to understand and explore the geometry of figures. It is incredibly useful for solving geometric problems algebraically, as it provides a clear framework to represent geometric figures as equations.
**Benefits and Uses**:
  • Combines the spatial visualization of geometry with the logical precision of algebra.
  • Allows complex geometry problems to be broken down into manageable algebraic tasks.
  • Facilitates understanding of distances, gradients, and relationships between geometric entities.
In doing so, Analytical Geometry provides the methodology to apply algebraic solutions to geometric problems, finding distances, angles, and other properties using formulas like the Point-to-Line Distance Formula.

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Most popular questions from this chapter

Given that \(k\) is a nonzero constant, which of the following are linear equations? a. \(k x-\frac{1}{k} y=3\) b. \(2 \sin x=k x\) c. \(2^{k} x+3 y-z=0\) d. \(\frac{1}{x}-y=3\)

A system of two equations in three unknowns has been manipulated, and, after correctly using elementary operations, a student arrives at the following equivalent system of equations: ? \(2 x-y+2 z=1\) ? \(0 x+0 y+0 z=0\) a. Write a solution to this system of equations, and explain what your solution means. b. Give an example of a system of equations that leads to your solution in part a.

The lines \(\vec{r}=(-3,8,1)+s(1,-1,1), s \in \mathbf{R},\) and \(\vec{r}=(1,4,2)+t(-3, k, 8), t \in \mathbf{R},\) intersect at a point. a. Determine the value of \(k\) b. What are the coordinates of the point of intersection?

Determine the solution to the following system of equations: ? \(\frac{1}{a}+\frac{1}{b}-\frac{1}{c}=0\) ? \(\frac{2}{a}+\frac{3}{b}+\frac{2}{c}=\frac{13}{6}\) ? \(\frac{4}{a}-\frac{2}{b}+\frac{3}{c}=\frac{5}{2}\)

Explain why there is no solution to the following system of equations: ? \(2 x+3 y-4 z=-5\) ? \(x-y+3 z=-201\) ? \(5 x-5 y+15 z=-1004\)
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