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

(a) Determine the equation of the tangent line to the circle $x^{2}+y^{2}=169$ at the point $(5,12) .$ (b) Compare the $y$ -values on the tangent line with those on the circle near $x=5$

$$\text { (a) } y=\frac{-5 }{ 12} x +\frac{169}{12}$$

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

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

05:31

(a) Determine the equation…

05:46

01:20

Tangent Line Find an equat…

01:09

(a) If $x^{3}+y^{3}-x y^{2…

03:57

Find an equation of the ta…

01:17

Find the equation of the t…

01:33

The length of the tangent …

06:10

For each implicitly define…

03:35

a. Find an equation for th…

for this problem, we have been given the equation of a circle X squared plus y squared equals 169. And we want to find the equation of the tangent line, uh, to the circle at are given point of 5 12. So let's just do a quick sketch so we can visualize what's going on. So we have a circle, and this 1 69 is the square of the radius. So the square root of 169 is 13. So this goes out 13 units all the way around. Now our 0.5 12 is gonna be No, let's say right about there. Okay, give or take again. It's it's just a sketch. But there's my point. My tangent line is going to be this green line right there. So one piece of information is important to know is a tangent line is always going to be, um, perfectly perpendicular to a radius drawn to the point where it touches the circle. So if I take a radius about to touch the tangent line, that's going to be perpendicular. So my tangent line, I'm just gonna call my tangent. I'm just gonna label this here. This is my tangent line. I have a point on it already. If I confined the slope of my tangent line, I can put this in point slope form and find the value of it. Well, what I can dio is I confined that the slope of that radius I confined this slope. That's going to be how I find the slope of the tangent because I confined the slope of that radius. Perpendicular lines always have, um, slopes that air negative, reciprocal of each other. So if I know the slope of that line with the radius that will give me the slope of my tangent line, I can write the equation. Okay, so let's look at the radius. What is the slope of that line? Well, I have two points. I know one end of that radius is at the origin and the other Is that the 10.5 12 so I can use my formula for slope. Remember? Slope is rise overrun. Rise is my change. And why? How fast I'm rising or falling so I can subtract. My wise run is the difference in my exes so I can subtract the exes. So I'm just gonna call this point here, 0.0.1. I'll call this 1.2. That's just the order I wrote them in. So why one minus y two over X one minus X two. So the slope of the line with that radius is 12/5. In order to find the tangent line slope, I have to take the negative. Reciprocal. So there's my slope. So my equation from for the tangent line is going to be in Point Slope form. Why I should let me write Put point Slope form first. Just we can see it. Why minus y one equals M times X minus X one x one y one There's my point and M is my slope. So I have why minus? Well, my point is 5 12. So it's Y minus 12 equals my slope Times X minus five. So let's get rid of thes parentheses. Here I have y minus 12 equals negative 5 12 X plus 25 12th, and let's add 12 to both sides. Removes it from the right hand side, and a common denominator would mean I'm adding 144 12th to the right hand to the right hand side. So that gives me why equals negative 5 12 x plus 169 12th. That is the equation of my tangent line. Now that we have that, the second part of this problem says, we just want to compare some of these Y values with the points that are on the tangent line and to the circle. What are the values of those? So what I've done is I've put together a spreadsheet. I've got the value of X at our point, which is five, and I've looked at some points that are a little bit bigger and some points that are a little bit smaller. And I've compared what the values of the why would be if I used the equation of my circle, which have circled in blue here. That's what we were given or the equation of the line. So the very top line here you can see that is there's no difference because it's tangent right at that point, the circle in the line of the same. I've gone up to six and down to four, and you can see that, especially just a couple 10th away. There's very little difference, and even up to one whole unit away. There's not a huge difference yet, but the farther away from 10.5 executing five you go, the bigger the difference. And I put that difference column just You can really see it again. It's not a huge difference yet, but every step I take away from that point where they touch the difference is a little bit bigger, every single step. So that's how the circle and the line values compare for our circle X squared plus y squared equals 1 69 and its tangent line.

View More Answers From This Book

Find Another Textbook

Numerade Educator

01:19

Refer to Figures $8,9,$ and $10 .$ In each case, choose another point on the…

05:28

(a) Find the slope of the line whose equation is $3 x+7 y+42=0$. Find the $x…

01:35

Determine whether the given equations is a circle, a point, or a contradicti…

01:02

Determine if the graph of the given equation is symmetric with respect to th…

04:01

The divorce rate (number of divorces per thousand couples) in the United Sta…

04:06

Find the indicated limit.Let $f(x)=\frac{x^{2}-4}{x+2},$ determine each …

01:55

(a) Determine if the given equation is a supply or demand equation. For a su…

sketch the graph of the given ellipse, labeling all intercepts.$$\begin{…

01:22

Determine the horizontal asymptotes, if they exists.$$f(x)=\frac{2 x^{2}…

04:47

Solve each of the quadratics by first completing the square. When the roots …