 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 exercises 1 to 6, find the coordinates of the fukai and the vertices, the intense eccentricity and the length of the latest rectum of the hyperbolas. Now, the equation which is given to us is x square by 16 minus y square by 9 is equal to 1. Before proceeding on with the solution, we should be well versed with what a hyperbola is all about. A hyperbola is the set of all points in a plane. The difference of whose distances from two fixed points in the plane is a constant. Now, here we have this as a standard equation of a hyperbola which is x square by a square minus y square by b square is equal to 1. Here, these two f1 and f2 are the fixed points which are called the fukai. The line through this fukai is called the transverse axis. The line through the center and perpendicular to the transverse axis is called the conjugate axis. The point of intersection of parabola with the transverse axis is called the vertices. Now, in the figure given here, hyperbola is shown, fukai are on x axis with f1 and f2 having coordinate plus minus c comma 0 and the vertices that is plus minus a comma 0. The general equation has been discussed with you before that is x square by a square minus y square by b square is equal to 1. Also, the eccentricity that is e is equal to c by a whereas we can find the length of the latest rectum is equal to 2 b square by a. Now, here the given equation of the hyperbola is this is the given equation of hyperbola. x square by 16 minus y square by 9 is equal to 1. Now, on comparing it with standard equation that is x square by a square minus y square by b square is equal to 1. We have a square equals to 16 that means a is equal to plus minus 4 whereas b square is equal to 9 that is b is equal to plus minus 3. Now, we know that we can obtain the value of b that is the square root of c square minus a square. Now, we know the value of b and a so with this the help of this equation we can find out the value of c that implies we will take the positive values only 3 is equal to c square minus a square is 16 on squaring both the sides we have 9 is equal to c square minus 16 that further implies 9 plus 16 is equal to c square that is 25 is equal to c square that further implies the value of c is plus minus 5. Now, since we know the value of a b and c thus we can see that we know that foci is plus minus c comma 0 right. So, we have it as its value is equal to plus minus 5 since c is plus minus 5 comma 0 vertices are plus minus a comma 0 and the value of a known to us is equal to 4. Now, eccentricity that is e is equal to c upon a that is 5 upon 4 and further we have the length of latest rectum is equal to 2 b square b is 3 by a that is 4. So, we have the answer as 9 by 2. So, let us write down the answers once again we obtained the value of foci equal to plus minus 5 comma 0 vertices are plus minus 4 comma 0 that eccentricity is 5 by 4 whereas length of latest rectum is 9 by 2. So, these are the required answers to the question that is given to you. Hope you enjoyed the first session and understood the concept of hyperbola as well as the standard of equation of the hyperbola well. Have a very nice day ahead.