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

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

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

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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).

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for this problem, we've 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 to this circle at the 0.12 negative five. So to begin this, let's take a quick look at what we're doing is it's just a real rough sketch. I have a circle. It is centered at the origin because it's just X squared and y squared R squared is 1 69 So the square root of that is 13. That means my radius for my circle is 13 and I have a point on my circle, which is 12 negative five. So let's say that's right about there. 12 Negative five and I want to draw a tangent line to that point roughly there. I want to find the equation of that tangent line Now, since I already have a point on that line, if I could find the slope of it, I could use the point slope form to find the equation of this line. And remember, Point Slope form is why minus why one equals m times X minus X one. Now I already have X one y one. That's my point. I just need the slope. Well, I don't have the slope, and I don't have another point on this line to get the slope. But what we do have we have a radius for our circle. And if you draw any circle, if you draw a radius from the center of the circle out to a tangent line, the radius and the tangent line will meet at a right angle. So these two lines are going to be perpendicular to each other. So if I confined the slope of the radius, I could take the negative reciprocal that will give me the slip of the tangent line. And I can find the slope of the radius because I have two points. I have the 20.12 negative five. That's our point that we've been given. And I have the center of the circle. That's the other end of the radius, so that 00 so all I have to do is find this slope. So remember, slope equals rise over run rise is how fast I'm going up or down. So that's my change in wise and run is how fast I'm going side to side. That's change in X. So for my two points here, I have negative five minus zero. That's my change in Wise and 12 minus zero. That's my change in X. So that gives me a slope for the radius of negative 5/12. To get the slope of my red tangent line, I take the negative, reciprocal positive 12 5th. So now I have my slope. I have my point. I confined the equation of my tangent line. Why minus negative? Five positive That's will be white plus five equals M 12 5th So X minus my X one, which is 12. So let's just clean this up a little bit. Let's put this into slope intercept form. We'll get rid of our parentheses. So that's 12 5th X minus 144 5th, and I'm going to subtract five from both sides. And when I do, we'll do a common denominator off. Five that B minus 25 5th, which gives me an equation for my tangent line of 12 50 x minus 169 5th. That is the equation of my tangent line. Now, the last part of this problem we want to compare, um, points on the line versus points on the circle. So what we're going to do is we know that the X coordinate, I'm just gonna put a little blue arrow there. The X coordinate is 12. That's where they meet. So I'm going to go a little bit bigger than 12 and a little bit smaller than 12. And I'm going to compare the Y values for the line in the circle. And I've done that in a spreadsheet that I'll show you in a moment. But first the line is already solved for why that's in slope intercept form. Let's take our circle and solve it for why that gives me why squared equals 169 minus X squared. And then I take the square root, which gives me plus or minus 169 minus X squared. Now, if you look at our circle here we are down in the fourth quadrant, the bottom half of the circle. That means I don't have to do just plus or minus. I'm just looking at the minus the plus values. Give me the upper half of the circle. The negative values give me the lower half. So for this case, I am in the lower half. I'll use the negative. So ah, plug in different values for X. And I could compare the blue equation, which is my circle and the red equation, which is my line. Okay, So as you can see, the first line on this spreadsheet has no difference. That's where they touch. I'm going up by 10 all the way up to 13, and it actually gets very far away as I get up a zai approach. 13. There's a huge difference. Uh, as I go down toward from 12 down to 11. The difference isn't as big, but it's certainly there. And as you can see from both sides, every step that I take farther away from where they touch the difference gets bigger and bigger and bigger. So the farther apart you are from from that point of intersection, the bigger the difference between my wife values for my circle, which was this equation and my line, which was this equation

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