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Determine whether the given equations is a circle, a point, or a contradiction (no real graph).$$x^{2}+y^{2}-10 x+4 y+29=0$$

Point

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

McMaster University

Harvey Mudd College

Baylor University

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).

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if this problem were examining the equation. X squared plus y squared minus 10 X plus four y plus 29 equals zero. And our goal for this problem is to identify what the graph looks like. Is it a circle? Is it a point, or is it a contradiction? In other words, if it's a contradiction, that means there's no graph on the real numbers that will illustrate this particular equation. So how do you know which of these three it is? What are we looking for? Well, to answer that, let's take a step back and look at the standard form of a circle. Standard form of a circle is X minus H squared. Plus why minus k squared equals R squared. OK, what we're gonna look for if we can put this into standard form, we're gonna look at that r squared. If r squared is any positive number, I have a circle. I would just take the square root of that number, and that tells me the radius of my circle if r squared equals zero, then I have a point. My circle exists at that center point HK, but it doesn't go anywhere. There's no radius so R squared. Equaling zero means that my circle has, um, shrunk down until it's on Lee a single point. And if r squared is negative, that's a contradiction toe. Have a circle. I have to have a radius, and it's a distance that that would be a positive number. Squaring a positive number gives me a positive number. Have our square to be negative. There is no real number radius work that will satisfy that condition. Therefore, the graph is a contradiction. So if I can have my circle in standard form, this is an easy problem to solve. Unfortunately, we have not been given this circle in standard form. This is the general form. So that's how we're going to approach this problem. We're gonna take the general form and then change it to standard forms that we can answer the question. To do that, we're going to need to complete the square twice. So I'm going to gather all my ex terms going to gather my white terms. And in that constant 29 I'm going to subtract from both sides, put it on the other side of my equation. Okay, Now we're going to complete the square to do that. We look at the in this case, the excess first. We'll look at the X term. We'll take half of that coefficient and then we square it, which means I'll be adding 25 I have to do that to both sides. So that's going to give me X minus five squared. Okay, Now, for our wise again, we're gonna look at the Y term, take half of the coefficient, and then square it. We have to add it to both sides, and that gives us why. Plus two squared. And if I add up all of my numbers on the right hand side, that gives me zero. Well, take a look at our three cases. That is the second option where r squared equals zero. That means for this particular equation, I have a point in particular. It is the center of this circle equation, which is five negative too

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