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

Question

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

$\frac{x^2}{4}-\frac{y^2}{16}=1$

Accepted Answer

Hello. Today in this video, we are going to be determining the graph that the equation represents the equation that is given to us is X squared divided by 28 minus Y squared, divided by 112 is equal to one. Now, in order to approach this problem, the first thing that we can do is we can go ahead and rewrite this conic into the general conic. In order to do this, we're going to want to rewrite this in the standard form that is equal to zero. So the first thing we want to do is we want to eliminate the denominators of the X and Y term. But in order to do this, we'll need to first figure out the common multiple between the two denominators. We are given 28 and 112. In the current denominators and the common multiple will be 112. In order to rewrite the X term to have a denominator of 112 we will first multiply the X term by four. Doing this will allow us to rewrite the equation as four X squared divided by 112 minus Y squared divided by 112 is equal to one next. In order to eliminate the denominators of 112 we will multiply the entirety of the equation by 112. This will allow us to simplify the equation to be four X squared minus Y squared is equal to 112. Finally, we want to set the equation equal to zero. So we are going to subtract both sides of the equation by 112. And this will allow us to rewrite the equation to be four X squared minus Y squared minus 112 is equal to zero. Now that we have the equation in standard form, we can use the general conic to help us identify which conic the equation represents. Now recall that the general conic is given to us as A X squared plus CY squared plus DX plus ey plus F is equal to zero. There are four conditions that follow the general conic. The first condition is that if the coefficients of A and C are equal to each other, then the conic represents a circle if the coefficients or the product of the coefficients A and C is equal to zero. This conic represents a parabola if the coefficients of A and C are not equal to each other and their product is positive or greater than zero. This conic represents an ellipse. And finally, if the product of A and C are less than zero or a negative number. This conic represents a hyperbola. So using the general conic, we will first need to identify our coefficients of A and C. The coefficient of A is the coefficient in front of the X term, which in this case is going to be four. So we have A is equal to four. Next, the coefficient of C is in front of Y. So the coefficient in our equation is going to be negative one. So C is negative one, notice that the coefficients of A and C are not equal to each other. So this rules out the possibility of this being a circle. Let's go ahead and check the product of ANC A multiplied by C will equal to four, multiplied by negative one and four multiplied by negative one will give us a product of negative four. This is a negative number and negative numbers are less than zero. So this indicates that the equation represents a hyperbola. So the overall equation represents the hyperbolic conic. And what this means is that the solution to this problem is going to be a. So I hope this video helps you in understanding on how to approach this 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.
$\frac{x^2}{4}-\frac{y^2}{16}=1$

Key Concepts

Conic Sections

Standard Form of Hyperbola

Identifying Graph Types from Equations

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