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Determine whether the given equations is a circle, a point, or a contradiction (no real graph).$$(x+4)^{2}+(y-3)^{2}-9=0$$

Circle

Algebra

Chapter 1

Functions and their Applications

Section 5

The Circle

Functions

Campbell University

Harvey Mudd College

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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for this problem, we're going to examine the equation. X plus four squared. Plus why minus three squared minus nine equals zero. And our goal is to see what this, uh, graph of this equation is. Is it a circle? Is it a point or is it a contradiction? In other words, there is no real graph that matches this equation. Well, how do we know which case we have to answer? That we're going to need to put this into standard form. Standard form for a circle is X minus H squared. Plus why minus k squared equals R squared. And this r squared is going to be what is going to be the thing that tells us which case we have. If R squared is greater than zero, we have a circle because that means I have some radius. My radius could be two or three or four, but I'm gonna have a radius if r squared equal zero. That's a point. I exist at the center point of my circle at the point HK. But I don't go anywhere. There's no radius, so that gives me a point. And if r squared is negative, that is a contradiction because there is no real radius. When I square it, that gives me a negative number. So I'm gonna take my equation we've been given, put it into standard form. And from that I'll be able to use my radius to find out what I actually have from my graph. So for this particular equation, I'm gonna be moving that negative nine, adding it. So my r squared is going to be nine positive value. I have a circle. In fact, I have a circle with a radius of three.

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