Find the standard form of the equation of the hyperbola satisfyin...

Question

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (-8,0), (8,0); Vertices: (-3,0), (3,0)

Accepted Answer

Hello. Today we're going to be finding the equation of the standard form of the hyperbole to Using the given information. Now, before we jump into this, let's just go ahead and quickly recall the equations of a hyperbole to. The first one is x squared over a squared plus, Y squared over B squared is equal to one. And the second form of this is y squared over a squared plus X squared over B squared is equal to one. Now the difference between this too is that the first option is when the X axis is the dominant axis or the transverse axis. And the second option here where y squared goes first is when the y axis is the dominant or transverse axis. So it really just depends on what is being given to us. So let's just go ahead and take a look at the given information. We are told that the focus, I are located at zero comma plus or minus five and we are told that the furnaces are located at zero comma plus or minus one. So let's just go ahead and graph what is given to us. So again, we are told that the fosse are located at zero plus or minus five and that means that they're going to be located along the Y axis one. Fosse at zero comma five and the other fosse at zero comma negative five. And we are told that the first set of vertex is are at zero comma plus or minus one. So we have to given versus is that both lie on the y axis, one at zero comma one And the other at 0:00 -1. Now notice that the photos I lie along the y axis. That means that since the photos are located on the y axis, the y axis is going to be the dominant or transverse axis. So the standard form of our equation is going to be the second form Y squared over A squared plus X squared over B squared is equal to one. Now we can represent the foe, see by the letter C. And we can represent the first set of vortices as a. So we want a squared and B squared. Since we are given the value of A. Which can be represented by one, squaring both sides of this equation is going to give us a squared is equal to one squared which is just equal to one and that's gonna be our first substitution. But now we need to find what B squared is. But we have an equation for hyperbole which states that B squared is equal to c squared minus a squared. In order to solve for b squared we need both. C squared is a squared. But recall that C which is the focus, i is equal to plus or minus five. So C is going to equal to five. If we square both sides of this equation, we get c squared is equal to five squared which is equal to 25 we can go ahead and use that as a substitution to solve for B squared. So b squared is going to equal to c squared which is minus a squared which is one and 25 minus one is going to give us 24. So we have B squared is equal to 24. And that's gonna be the substitution. We need for our general equation for the hyperbole to So again we have y squared over A squared plus X squared over B squared is equal to one. Now again, we have our substitution for a squared earlier we said that a squared is equal to one. So we're just gonna go ahead and substitute A squared for one and we just sold for B squared. We said that b squared is equal to 24. So we can replace B squared with 24. And that means that the equation of this hyperbole to is going to be y squared over one plus X squared over 24 is equal to one. And that means that the answer to this problem is going to be a So I hope this video helps you in understanding how to find the equation of a hyperbole to. And I'll go ahead and see you all in the next video

Find the standard form of the equation of the hyperbola satisfying the given conditions. Foci: (-8,0), (8,0); Vertices: (-3,0), (3,0)

Key Concepts

Hyperbola Definition

Foci and Vertices

Standard Form of Hyperbola Equation

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