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Determine whether the given equations is a circle, a point, or a contradiction (no real graph).$$x^{2}+y^{2}+8 x+6 y+16=0$$

Circle

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

Missouri State University

Baylor 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).

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Determine whether the give…

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Decide whether each equati…

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So this problem we're going to examine the equation X squared plus y squared plus eight X plus six y plus 16 equals zero. And the goal of this exercise is to identify what the graph of this equation looks like. Is it a circle? Is it a point, or is it a contradiction? In other words, ah, contradiction means that there is no graph in the rial numbers that will, um, illustrates this particular equation. So how do you know, How do I know which one of these three cases I'm looking at? Well, to answer that, let's review what the standard form of a circle is. X minus h squared. Plus why minus k squared equals R squared. So R squared is the radius of our circle. That's the piece I'm going to be looking at. If r squared, it's positive. Any positive number? I have a circle. All I would do is take the square root of that number that gives me the radius. And then, if I have the center Aiken plot my circle. So anti positive number for R squared is a circle. If r squared equal zero, that means I have a point. Um I have the center of my circle HK, but then I can't go anywhere. I have no radius. So, um, if r squared to zero means my circle has shrunk down until it is a single point at the center. And if r squared is less than zero, that's my contradiction. R squared any real number when you square, it has to be positive if I end up with a negative there. That means that there that it is not a really number answer. There is no radius that will satisfy this equation. So that's how I'm going to determine whether this equation that I've been given is a circle a point or a contradiction. Now, unfortunately, this is general form that we've been given everything equal to zero. I want standard form, so that's our goal for this problem. We're going to go from general form to standard form and then we'll be able to answer. So standard form means I'm gonna have to complete the square twice once for my ex terms will collect those together once for my why terms. So I'll collect those together and that 16 that constant. I'm going to subtract from both sides will have have that equal to negative 16. Now let's complete the square. We'll start with the excess. First. We look at the X term, take half of the coefficient and square it. That's the number that were missing. And I could do that as long as we do it to both sides. So that gives me an X Plus four squared now for the wise again, we look at the Y term. We take half of the coefficient and we square that number. We have to add that then to both sides. So that gives us why. Plus three squared equaling nine. Well, let's take a look at that are squared position. That's a positive number. That's our first case. So for this equation, the graph is going to be a circle.

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