 Hello and how are you all doing today? My name is Priyanka and I shall be helping you with the following question. It says in each of the exercise 1 to 6, find the coordinates of the fukai and the vertices, the eccentricity and the length of the latest rectum of the hyperbola. Now here we are given the equation of the hyperbola as 16x square minus 9y square is equal to 576. Here we are given the equation of hyperbola as 16x square minus 9y square is equal to 576. Now dividing the whole equation by 576 we have x square by 36 minus y square by 64 is equal to 1. Now on comparing it with the general equation of hyperbola that is x square by a square minus y square by b square is equal to 1. We have x square by a square minus y square by b square is equal to 1. We have the value of a as 6 and the value of b as 8. Let us find out the value of c with given relation that we know that since b is equal to under the root c square minus a square we can obtain the value of b that is 8 is equal to c square minus a square is 36. Now squaring both the sides we have 64 is equal to c square minus 36 that further implies that 36 plus 64 is equal to c square that is 100 is equal to c square therefore c is equal to 10. Now we have foci as it is plus minus c comma 0 so it will be plus minus 10 comma 0 vertices are plus minus a comma 0. So we have plus minus 6 comma 0. Now the eccentricity that is e is c by a that is 10 by 6 that on simplifying can be written as 5 by 3. And lastly the length of latest rectum is equal to 2b square 2 into b square is 64 by that on simplifying can be written as 64 by 3. So these are the answers to the given question. Hope you understood the whole concept of hyperbola well. Do remember its standard equation and have a very nice day ahead.