 Hello and how are you all dear? My name is Priyanka and let us discuss this question it says in each of the following exercises 1, 2, 6, find the coordinates of the Foucaille and the vertices, the eccentricity and the length of latest rectum of the hyperbola. Now the standard equation, sorry the equation which is given to us is y square by 9 minus x square by 27 is equal to 1. Now let us first understand the definition of a hyperbola. A hyperbola is a set of all points in a plane the difference of whose distances from two fixed points which we called as Foucaille in the plane is a constant. Now we have the standard equation of hyperbola where the Foucaille are on y axis with Foucaille f1 and f2 this is f1 and this is f2 and vertices v1 and v2 that is 0 comma plus minus a and Foucaille are 0 comma plus minus c. Now the general equation of the hyperbola in this form is y square by a square minus x square by b square is equal to 1. Also we need to find out the Foucaille which we have explained to you as 0 comma plus minus c where vertices are 0 comma plus minus a eccentricity that is e is equal to c by a and the length of latest rectum is found by multiplying 2 with b square and dividing their product by a. Right now the here the general equation is given to us as y square by 9 minus x square by 27 is equal to 1. Now in comparing it standard equation which is discussed in this diagram which is y square by a square minus x square by b square is equal to 1. The value of a square as 9 that means a is equal to plus minus 3 and value of b square is equal to 27 that means we have b as plus minus 3 root 3. Right also we know that b square is equal to c square minus a square. Now we know b square as a square we can easily find out c square so on substituting the values we have b square as 27 c square is what we need to find out minus a square that further implies 36 is equal to c square that further implies c is equal to plus minus 6. Now in the question we were asked to find out for chi and we know that it is plus 0 comma plus minus c which is found to be 6 then we have vertices that is 0 comma plus minus a and that is found out to be 3 above further eccentricity we have c by a on simplifying we have the answer as 2 and lastly the length of latest rectum so we have the length of latest rectum as 2 into b square which is 27 by a that is found out to be 3 on simplifying we have the answer as 18 so the answer to the question is found out hope you understood the concept well and enjoyed the session too hope you have a good time ahead.