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 asked to figure out what the equation of a hyperbole is going to be with a given fosse advertises. So before we jump into this let's just go ahead and recall the possible forms of the equation of a hyperbole. So if our X axis ends up being our transverse axis then the form of that equation is going to be x squared over a squared minus Y squared over B squared is equal to one. And if we don't have the X axis as the transverse axis but rather the y axis is our transverse axis then the form of that equation is going to be y squared over a squared minus X squared over B squared is equal to one. So in order to figure out what equation we need to find, we need to figure out what form of the equation we're working with first. Well let's go ahead and graph are given fosse and versus is so the given fosse is at zero comma plus or minus five and the given vertex is is at zero comma plus or minus one. So if we were to quickly put this on an axis arf O C R at plus or minus five. So right here at zero comma five and here at zero comma negative five. So this is gonna be our c values and our versaces are at zero comma plus or minus one. So right right below zero comma five, that's gonna be our a value And right above 0:00 -5. And that's gonna be our other a value. So since all of our points lie on the y axis, that means that our transverse axis is gonna be the y axis. So the form of our equation is going to be y squared over a squared minus x squared over b squared equal to one. Okay, so since we are working with that, we need to figure out what our a squared and r b squared term is going to be. Well, we know that A is going to equal just based off the given vertex C. And that means that a squared is going to equal to one squared which is still one and we don't have a B term but we do have a C term which is going to be five based off of the given faux side. And in order to figure out what the B term is, we do have an equation that can help us out with that. And specifically for hyperbole, as we can go ahead and use b squared is equal to c squared minus a squared. So in order to use this equation to figure out what our b squared term is going to be, we need to first have a squared and c squared, we figure out what A squared is, but now we need a c squared. So squaring both sides of this equation here is going to give me c squared equal to five squared which is equal to 25. Now, what I'm gonna do is go ahead and take our c squared value and our a squared value and plug it into this equation to figure out what our b squared term is going to be. So B squared is going to equal to C squared, which is 25 minus a squared, which is one. That means that B squared is going to equal to 24. And now we have a value for a squared and B squared. And we just want to go ahead and plug that in to the equation of a hyper below where the transverse axis is the y axis. Doing so is going to give us y squared over A squared, which is one minus X squared over B squared, which is 24 equal to one. And that's gonna be the equation of our harper bola. That also 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

Watch next