Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Given the two parallel lines $y=m x+b$ and $y=m x+B,$ determine the perpendicular distance between these two lines.

$\frac{|B-b|}{\sqrt{m^{2}+1}}$

Algebra

Chapter 1

Functions and their Applications

Section 1

The Line

Functions

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

03:18

01:07

Express the perpendicular …

01:44

Use the facts that paralle…

01:16

The equations of two lines…

02:58

Determine whether the line…

for this problem we're going to take to generic parallel lines. Y equals MX plus lower case B y equals MX plus upper case be. And our goal is to determine the perpendicular distance between these two lines. So I'm just going to draw this. We could get a feel for what we're talking about. I am, uh, going to draw in. You know, since I'm doing a picture here on the grid, it's going to actually have real numbers, but I'm not going to reference them. I'm not gonna know if it's positive and negative. Not gonna physically count. This is just for reference. So I'm going to call this first line here. This is gonna be lying a That's that'd be my first one here. And the second one will be be okay. They have the same slope, the parallel lines, and I know there. Why intercepts. So we're going to call this, um, be so this point here is going to be zero little bee. And down here, this is going to be zero capital B. Again. Um, my picture has one positive and one negative. I'm not making any assumptions about which one's bigger. A smaller positive or negative. This is just for reference. Okay, What I want to do to find the perpendicular distance, I'm going to draw in the perpendicular line from one to the other. Okay? And I want this distance. I want the distance from where a crosses that blue perpendicular line to where be crosses that blooper perpendicular line. So I need those two points. Well, I've drawn the perpendicular line through the same. Why intercept his line A zero B. So that first intersection line I already know what I need to find out is that second one. So in order to do that, we're going to find the equation of the perpendicular line and then set that equal to B and see where they intersect. That will be my second point. Then I can use the distance formula to find the distance. Okay, so let's find I'm gonna call this see this perpendicular line? Let's find the equation of see what I know is it has the same. Why intercept? So I know it's gonna be a plus. Lower case B. And I know that it's a slope. Is the negative reciprocal. So the slope is gonna be negative one over M. So that's my equation. So let's set C and B equal to each other. What? What? X and y make that system of equations true. That will give me that second intersection point. So what I have I'm just gonna since they both equal toe why, I'm just gonna set both right hand sides equal to each other and I want to solve for X. So I'm going to add one over M x to both sides, and I'm going to subtract Capital B. Okay, factor out an X actually, no, actually, I'm not gonna be. In fact, I'm just gonna put these together. I'm gonna have a common denominator. That first term becomes m squared over em. So I m squared, plus one over m times X equals little B minus Bigbie. And to solve that, that multiplies by em on top. I have m squared plus one on the bottom. So that's the value of X. Um, the X coordinate of that intersection point. So what's the wipe intersection or what's What's the Y? Coordinate? Well, let's come up and let's use um See, I'm gonna plug this X into that third equation and solve for? Why? So why equals negative one over M times X, which is B minus B times M over M squared plus one plus b. Okay, No, let me see what cancels. I have an m on top M on bottom. That's great. And on top, I have a negative little B minus B over M squared plus one plus B. So it doesn't look very pretty, But these are the X, and why coordinates of that intersection of lines B and C. So now I want the distance between the 0.0 little B and the point that these coordinates represent, So let's begin that distance is the square root. First, I'm going to do the difference of my wise squared. So I have negative B minus B over m squared plus one plus B. That's the white quartet of the first piece, minus B, which is the white corner of the second piece squared. Now get the same thing for the axis. I have B minus B little B minus Bigby times M over M squared plus one squared. Well, fortunately, those bees cancel What I end up with is the square root okay on top when I square that top the negative will be gone so I'll just have little B minus big B squared over m squared plus one squared. And then over here, I have em squared over m squared, plus one squared. Well, fortunately, I have common denominator, and it is a perfect square, so I could just say the denominators m squared, plus one on top on top. I'm sorry. I'm gonna go back. I forgot to write this down. I have a little B minus, Big B squared. I can factor that out. And I'm left with one plus m squared and my very last little bit here. And I'm gonna come just to make sure I got enough for him yourself. A little bit more space on top. I have a square, so I'm gonna take the square root of little B minus big b squared. And I'm going to put that in absolute values because it's going to be a positive value. I'm taking the square root. So I'm not making any assumptions about which one's bigger or smaller Thea salute value of that distance and I'm gonna square root of one plus m squared. Since it's the same thing I have on the bottom. That's equivalent to having that m squared plus one in the denominator. So that is the perpendicular distance between my two lines.

View More Answers From This Book

Find Another Textbook

Numerade Educator

01:06

You are given a pair of supply and demand equations; identify which is suppl…

03:23

Use the first derivative to determine where the given function is increasing…

04:28

Suppose the power station in the previous exercise is moved one mile inland …

01:52

$$f(x)=\left\{\begin{aligned}4 x-2 & \text { if } x \leq 1 \\x+1 & \…

02:07

(a) Find the $x$ -intercept(s); (b) Find the vertical asymptotes; (c) Find t…

01:59

Find two functions $f$ and $g,$ whose composition $f(x)$ ) will result in th…

02:52

Find $\frac{d}{d x}\left(\frac{4 x^{6}+3 x^{3}-8 x}{6 x^{5}}\right)$ by: (a)…

02:10

Determine the equation of the tangent line to the given curve at the indicat…

03:24

Find the average velocity over the given time interval, if $s=f(t)$ is the e…

01:04

The maximum possible demand for a certain commodity is 20,000 tons. The high…