 Hello and how are you all today? My name is Priyanka and let us discuss this question. It says in each of the exercise 7 to 15 find the equation of the hyperbola satisfying the given condition. Foucaille is given to us as plus minus 3 root 5 comma 0 and the latest rectum is of length 8. So let us proceed on with our solution. Foucaille is given to us as plus minus 3 root 5 comma 0 and the length of the latest rectum is given to us as 8. Now we know that since the Foucaille lies on x axis therefore the standard equation of hyperbola is x square by a square x square by a square minus y square by b square is equal to 1. And here the Foucaille is given to us as plus minus 3 root 5 comma 0 which is also equal to plus minus c comma 0 and the latest rectum is equal to 2b square by a. So from this we can conclude that therefore the value of c is equal to 3 root 5. Also we have 8 is equal to 2b square by a that implies 8a by 2 is equal to b square that further implies b square is equal to 4. Now with the help of the value of c and b we can find out the value of a. We know that b square c square minus a square. So we have 4 is equal to c square will be 3 root 5 the whole square minus a square that is further equal to 4 is equal to sorry it was 4a. So we have 4a is equal to 45 minus 3 root 5. So we have 4a is equal to a square which is a square plus 4a minus 45 is equal to 0. On splitting the middle term we have a minus 5 and a plus 9 equal to 0. Thus we have the value of a equals to 5 or a equals to minus 9. Since a cannot be negative therefore we have the value of a as 5. So we have the value of b square. So b square will be equal to what that was 4a. So it will be equal to 4 into 5 that is equal to 20. Now we know the value of a as well as b square. So we can have the general equation of the hyperbola as x square by a square that will be equal to 25 minus y square by b square that is 20 and it is equal to 1. So this is the required answer to this section. Hope you enjoyed and understood it. Have a nice day ahead.