In Exercises 27–32, find the standard form of the equation of eac...

Question

In Exercises 27–32, find the standard form of the equation of each hyperbola.

Accepted Answer

Hey, everyone here we have a given graph of a hyperbola. And we are asked to determine the standard form of its equation. Here we have four answer choice options. Answer choice A is X squared divided by minus Y squared, divided by 25 is equal to one B X squared divided by 25 minus Y squared, divided by 16 is equal to one answer. C X squared divided by four minus Y squared, divided by five is equal to one answer choice D X squared divided by five minus Y squared divided by four is equal to one. So taking a look at our given graph here, the first thing we want to observe is the center of our hyperbola. And here we see that the center is at the origin. So this tells us that the center is the 0.0. And since we know the center is the origin, we can recall the standard form of the equation of a hyperbola whose center is at the origin. And so we will have two formulas, one of which is X squared divided by A squared minus Y squared, divided by B squared is equal to one or we have Y squared divided by a squared minus X squared divided by B squared is equal to one. Now, we want to observe the vertices that we have from our graph and we see that we have our vertices at negative five and zero and the other vertices is at five and zero. And so, since the vertices lie on the X axis, then the transverse axis lies also on the X axis. And so this tells us that the equation of the hyperbole that matches our graph is this first formula that we have, which is X squared divided by A squared minus Y squared, divided by B squared is equal to one. So now we have identified the formula that we are going to need to use. And so now again, from our vertices which are negative 50 and five and zero, this tells us that our value for A is going to be five. And now we want to observe that the rectangle Passes through the points zero and 4 and zero and -4. And here we see from these points that we are four units above and four units below the center with these points. And therefore we know that our value for B is going to be four. So now we have identified the necessary formula that matches our graph as well as the values for A and B. So all that is left to do is substitute in A and B into this formula to get our equation. So we have X squared divided by five squared minus Y squared divided by four squared is equal to one. And now we just want to simplify this equation by squaring the values in the denominator. And so simplifying we have our final equation as X squared divided by 25 minus Y squared divided by 16 is equal to one. And so with our final equation here, we see we are left with answer choice B where again our equation of the hyperbola which matches the graph shown is X squared divided by 25 minus Y squared, divided by 16 is equal to one. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

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In Exercises 27–32, find the standard form of the equation of each hyperbola.

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