 Hello and how are you all today? My name is Priyanka and I shall be helping you with the following question. It says in each of the following exercise This is 7 to 15 find the equation of the hyperbola Satisfying the given conditions. Now the condition which is given to us is what this is is 0 comma plus minus 5 and 4 chi is given to us as 0 comma plus minus 8 Now we should be well versed With the standard equation of hyperbola when the foci lies on y-axis And that is y square by a square minus x square by b square is equal to 1 So to find out the standard equation of the hyperbola. We should be well versed with the value of a and b Also, since we know the foci and the vertices that means the value of c and a So we can easily find out the value of b With the known formula that is b square is equal to c square minus a square. So let us proceed on with the given solution Now here we are given a hyperbola with vertices 0 comma plus minus 5 foci as 0 comma plus minus 8 Now the vertices and the foci lies on y-axis Thus we have hyperbola of general equation that is y square By a square minus x square by b square is equal to 1 right further We know that here in the hyperbola of general equation of this type we have Vertices as 0 comma plus minus 8 and Here Vertices are 0 comma plus minus 5 So that means the value of a is equal to what exactly it's equal to 5 further foci is 0 comma plus minus c and here We are given foci as 0 comma plus minus 8 So that means the value of c is what exactly 8 Also, we know that b square c square minus a square. Let us quickly find out the value of b. It will be c square Minus a square we have c as 8 and a as 5 Which is b square is equal to 64 minus 25 That is equal to 39 that further implies that the value of b is root 39 So we have the equation of hyperbola y square by a Square that is 5 square minus x square by b square that is root 39 the whole square is equal to 1 or We have the answer as Y square by 25 that is 5 square minus x square by 39 is equal to 1 and this is the required equation of the hyperbola That was needed to be found out in the question. So to solve this question We use the given conditions and found out the equation of the hyperbola This ends my session. Hope you enjoyed and understood but have a very nice day ahead