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Determine whether the given equations is a circle, a point, or a contradiction (no real graph).$$(x-1)^{2}+(y+2)^{2}+4=0$$

Contradiction

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

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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're going to examine the equation. X minus one squared plus y plus two squared plus four equals zero. And we want to know what the graph of this looks like and we have three possibilities. First, it could be a circle. Second, it could be a point, or it could be a contradiction. It's possible that there is no graph that I can draw a real graph that satisfies this equation. Well, how do we know which one is which to look at that? Let's examine the standard form for a circle X minus h squared. Plus why minus k squared equals R squared. So if we can put the equation into this format, R squared will tell us what this looks like. If R squared is positive, bigger than zero, then I have a circle that we have radius. So if it's nine, I have a radius of three. If it's 16, I have a radius of four. If it's zero, that means I have a point. My radius zero I I exist, but I don't go anywhere, so it's just gonna be the graph of a point. And if it's negative, that's a contradiction There is no real number that Aiken Square to get a negative, so I can't draw this on a real graph. So let's take a look at our given equation and see which of these three cases we have to fully set this into our standard form. I'm gonna move that constant to the right hand side. So I have X minus one squared plus y plus two squared equals negative four. Well, a negative. That's our last case. So this there is no real graph for this. This is a contradiction.

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