 Hello and welcome to the session. The question says in each of the exercises one to find, find the coordinates of the foci, the vertices, the length of the major axis, the minor axis, the eccentricity and the length of latest rectum of the ellipse. Seventh is 36x square plus 4y square is equal to 144. So let's start with the solution and here we are given the equation of the ellipse. 36x square plus 4y square is equal to 144 or dividing both side by 144 we have 36x square upon 144 plus 4 upon 144y square is equal to 1 or 36x into 4 is 144 and 4 into 36 is 144 so we have x square upon 4 plus y square upon 36 is equal to 1 or it can further be written as x square upon 2 square plus y square upon 6 square is equal to 1. An ellipse is the set of all the points in a plane. The sum of force distances from two fixed points in a plane is constant. So here as we can see the denominator of y square upon 36 is greater than the denominator of x square upon 4. Therefore the major axis along the y axis and the standard form of the equation of ellipse is given by x square upon b square plus y square upon a square is equal to 1. If the major axis is along the y axis and on comparing the given equation with the standard form of the equation we find here that a is equal to 6 and b is equal to 2. Now please refer to question number 2 of this exercise to learn the standard form of equation of a ellipse and the major axis is along the y axis and how do we find the four kind vertices length of major axis, length of minor axis, eccentricity and latest rectum. Now from there we can see that c is equal to root over a square minus b square and let us find the value of c. So this gives 36 minus 4 is equal to root over 32 or this can also be written as 4 root 2. Now let us find the foci which is given by 0 comma plus minus c. So we have 0 comma plus minus 4 root 2 and we have to find the vertices and the coordinates of the vertices are given by 0 comma plus minus a and here a is equal to 6 therefore we have 0 comma plus minus 6. Now let us find the length of major axis given by 2 times of a. So we have 2 into 6 that is equal to 12 and similarly let us find the length of minor axis which is given by 2 into b. So we have 2 into 2 that is equal to 4. Now let us find the eccentricity denoting it by small e it is equal to c upon a and here c is 4 root 2 and a is 6. So we have 2 root 2 upon 3 as the eccentricity and lastly let us find the length of latest rectum and this is given by 2 into b square upon a so we have 2 into bs2 so we have 2 square upon 6 2 into 3 is 6 so the latest rectum is equal to 4 upon 3. Hence our answer is first we have to find the foci so foci are 0 plus minus 4 root 2 and the vertices of the given equation of the ellipse are given by 0 plus minus 6 the length of the major axis is equal to 12 the length of the minor axis is equal to 4 eccentricity which is denoted by small e is equal to 2 root 2 upon 3 and the length of the latest rectum is equal to 4 upon 3. So this completes the session hope you have understood it bye and take care.