Without actually graphing, identify the type of graph that eac...

Question

Without actually graphing, identify the type of graph that each equation has.

$\frac{\left(x+3\right)^2}{16}+\frac{\left(y-2\right)^2}{16}=1$

Accepted Answer

Hello, today we are going to be determining the graph that the equation represents the equation given to us is X plus 24 squared divided by 72 plus Y minus 37 squared divided by 72 is equal to one. One approach that we can use in order to determine the shape of the graph is to use the general conic. Now, the general conic is given to us as A X squared plus CY squared plus DX plus EY plus F is equal to zero. With this general conic, there are four conditions that follow it. The first condition states that if the coefficients of A and C are equal to each other, then this represents a circle. If the product of A and C is equal to zero, then the equation represents a parabola. If the coefficients A and C are not equal to each other and their product is greater than zero, then this represents an ellipse. And if the product of A and C is less than zero or in other words, a negative number, the equation represents a hyperbola. So in order to use the generic constant or the generic conic, we will first need to expand the given equation and set it equal to zero. In order to do this, we will first need to eliminate the denominators of the equation. Notice that both of the denominators are the same. They are both 72. In order to eliminate these denominators, we can multiply the entirety of the equation by 72. This will allow us to rewrite the equation as X plus 24 squared plus Y minus 37 squared is equal to 72. The next thing we will need to do is we will need to expand the groups of X plus 24 squared and Y minus 37 squared, X plus 24 squared is the same thing as saying X plus 24 multiplied by X plus 24. In order to expand this group, we will need to foil the two groups together and foiling X plus 24 with itself will give us X squared plus 48 X plus 576. If we take a look at Y minus 37 squared, Y minus 37 squared is the same thing as saying, Y minus 37 multiplied by itself. So just like the previous group, we're going to need to foil these quantities together and foiling these groups together will leave us with Y squared minus 74 Y plus 1369. So if we add these two groups together, we're going to end up with quite the lengthy equation, we will have X squared plus 48 X plus 576 plus Y squared minus 74 Y plus 1369 is equal to 72. We can go ahead and combine like terms together. The terms that we can combine together is 1369 and 576 combining these terms together and reordering some of the terms will leave us with X squared plus Y squared plus 48 X minus 74 Y plus 1945 is equal to 72. Finally, in order to set this equation equal to zero, we will subtract 72 from both sides of the equation. This will leave us with X squared plus Y squared plus 48 X minus 74 Y plus 1872 is equal to zero. Now, what we have here is the general form of a conic. Now again, in order to use the general form of the conic, we will need to identify our A and C coefficients. The A coefficient comes from the X squared term. And in our equation, X squared has a coefficient of one. So we're going to be setting a equal to one. Our C coefficient comes from the Y squared term and Y squared has a coefficient of one as well. So C is going to equal to one notice that the values of A and C are similar or exactly the same, which are both one, this implies that the equation represents the circle conic. And with that being said, the solution to this problem is going to be B so I hope this video helps you in understanding on how to approach this type of problem. And I'll go ahead and see you all in the next video.

Without actually graphing, identify the type of graph that each equation has.
$\frac{\left(x+3\right)^2}{16}+\frac{\left(y-2\right)^2}{16}=1$

Key Concepts

Conic Sections

Standard Form of an Ellipse

Graphing Techniques

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