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: (0,-4), (0,4); Vertices: (0, -2), (0,2)

Accepted Answer

Hello today we're going to be determined. The standard form of an equation of a hyperbole to with our given fossa and versaces. So let's just quickly recall what the possible forms of the equation of what hyperbole are. If our X axis is our transverse axis then the form of that equation is going to be x squared over a squared minus Y squared over B squared equal to one. And if our 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 the equation of the hyperbole is going to be, we need to first figure out what our transverse axis is going to be. And in order to do that we can just quickly graph what is given to us. So our focus I are located at plus or minus the square root of 61 comma zero. So just quickly graphing that we have fosse at square root negative square root 61 comma zero. So that's a C. Value and a positive square root 61 comma zero. And our vertex is are located at negative five comma zero. So in a value and positive five comma zero. Our other a value. So since all of our points are located on the X axis that means that the transverse axis for this hyperbole to is going to be the X axis. So this is gonna be the form of the hyperbole to that we're gonna want. Now the only thing we're missing is our A squared and B squared terms. Well, our versus is normally represent the eight term. So we can go ahead and let a Equal to the value of the versaces, which is going to be five. And since we want a squared squaring both sides of this is going to give me a squared is equal to five squared which is equal to 25 and we need to figure out a B term which we don't have but we do have an equation that can help us with that. And that equation is going to be, B squared is equal to c squared minus a squared. So in order to figure out what B squared is, we have an a square term but we need our c squared term, luckily we do have a C term which is going to be our fosse. So C is going to equal to the square root of 61 and squaring both sides of this is going to give me C squared is equal to the square root of 61 squared. Now the square cancels the square root leaving us with just 61. So c squared is equal to 61 a squared is equal to 25. We can go ahead and take those two values and plug it into this equation to get our B squared turn doing so is going to give me b squared is equal to c squared which is 61 minus a squared, which is 25 61 minus 25 is going to give me 36 so b squared is equal to 36. So now we have a value for B squared and we have a value for a squared. So let's go ahead and plug it into the form of a hyperbole to where the X axis is the transverse axis. Doing so is going to give me X squared Over a squared, which is 25 minus y squared over B squared, which is 36 equal to one. And that's going to be the standard form of the equation of our hyperbole. And 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 and I'll go ahead and see you all in the next video.

Watch next