Find the standard form of the equation of the hyperbola with vert...

Question

Find the standard form of the equation of the hyperbola with vertices (0, −6) and (0, 6), passing through (0, 9).

Accepted Answer

Hey, everyone. Here we are asked to work out the standard form of the Hyperbola equation. Given the following conditions here, the vertices are zero negative nine and 09 and the hyperbola passes through the 90.96. Here we have four answer choice options. First answer choice A X squared divided by minus Y squared divided by 81 is equal to one answer. Choice B Y squared divided by 18 minus X squared divided by 81 is equal to one answer choice C Y squared divided by 18 plus X squared divided by 81 is equal to one. And lastly answer choice D X squared divided by minus Y squared divided by nine is equal to one. So our first step here is to recall the equation of a, of the hyperbola with center at H K and axis parallel to the Y axis two. A is of the form the quantity of Y minus K squared all divided by A squared minus the quantity of X minus H squared, all divided by B squared. And this is equal to one. So now we want to begin by finding the center of our hyperbola and we can do this using our given vertices. So again, our given vertices are zero and negative nine and the 90.0 and nine. So to find the center, we just need to sum the X coordinates and divide them by two. And then sum the Y coordinates and then again, divide the sum by two. So since both of our X coordinates are zero, our overall X coordinate for the center is just going to be zero. And now for the Y coordinate, we have negative nine plus nine all divided by two. So now simplifying we have zero and then negative nine plus nine gives us zero. So we have zero divided by two, which is just zero. So we see our center is going to be at the origin. So now we want to find the distance between the two vertices so that we can find our variable B from our formula. So finding the distance between two vertices, we have two B is equal to negative nine minus nine. And now simplifying we get negative 18 here. So we have two B is equal to negative 18. Well, we can just take the absolute value of negative 18 because we are just mes measuring distance. So we actually just have two B is equal to 18. And so now just solving four B by dividing both sides by two, we see that our value for B is just nine. So now with this value for B identified, we can rewrite our formula to now match our current information. So we have our current formula as Y squared divided by A squared minus X squared divided by nine squared is equal to one. And now we also have given that the hyperbola passes through the 0.96. So to find our value for A, we just need to use this point of which the hyperbola passes through by plugging in this these values into our equation. So plugging in these values here, we have six squared divided by A squared minus nine squared, divided by nine squared is equal to one. And so now just solving for a, we can first simplify. So we have six squared divided by A squared minus one is equal to one. Now adding one to both sides, we have six squared Divided by a squared is equal to two. And now Rearranging our equation, we have six squared divided by two is equal to a squared. So we have a squared is equal to 18. And now taking the square root of both sides, we get a is equal to plus or minus the square root of 18. And simplifying even further, we have a is equal to plus or minus three times the square root of two. So now here we have a positive in negative value for a but we only need to take this positive value since we are taking the square of this value anyway, as shown in our equation. So now we have our value for A and our value for B identified. So we can just plug in these values into our equation. So we have Y squared divided by the quantity of three times the square root of two squared minus X squared divided by nine squared is equal to one. Now, we just want to square the terms in the denominator to simplify our equation. So simplifying we have Y squared divided by 18 minus X squared, divided by 81 is equal to one. So now we have found our equation for our hyperbola. So comparing our equation with our answer choice options, we see we are left with answer choice. B thanks for watching. I hope you found this video helpful and I'll see you again next time.

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Find the standard form of the equation of the hyperbola with vertices (0, −6) and (0, 6), passing through (0, 9).

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