 Hello and welcome to the session, my name is Asha and I shall be helping you with the following question which says, in each of the following exercises 10 to 20, find the equation for the ellipse that satisfies the given condition 12th one is vertices plus minus 6 comma 0 and 4 chi plus minus 4 comma 0. Now an ellipse is the set of all the points in the plane, the sum of whose distances from two fixed points in a plane is constant, therefore first let us draw an ellipse whose major axis is along the x axis. Suppose we represent the ellipse like this, here its major axis is along the x axis and some parameters of the ellipse are also shown, f1 and f2 are the two focus, line a b is the major axis and c d is the minor axis, a is the length of the semi major axis therefore 2a is equal to the length of major axis and b is the length of semi minor axis therefore 2b is equal to the length of minor axis and c which is the distance of focus from the center is given by root over a square minus b square and the standard form of equation of an ellipse whose major axis is along the x axis is given by x square upon a square plus y square upon b square is equal to 1 and the vertices of this ellipse are given by plus minus a comma 0 and the foci of this ellipse is given by plus minus c comma 0. With the help of these few ideas we are going to find the equation of the ellipse so this is our key idea. Now let us start with the solution, here we are given the vertices plus minus 6 comma 0 and the foci this plus minus 4 comma 0. Now as we can see the vertices as well as the foci lie on the x axis since the y coordinate is 0 in both the cases thus we have an ellipse whose major axis is along the x axis as we have shown in the key idea. Let us write down since the vertices are on the x axis the equation will be of the form x square upon a square plus y square upon b square is equal to 1 where a is the length of the semi major axis and since we have vertices of this equation as plus minus a comma 0 therefore on comparing it with the vertices plus minus 6 comma 0 we find that a is equal to 6 and foci is given by plus minus c comma 0 thus on comparing it with the given foci plus minus 4 comma 0 we get that c is equal to 4. Now let us find the value of b and since we have c which is the distance of the focus from the center which is given by a square minus b square so now let us from here find the value of b square substituting the value of c and a we have 4 is equal to root over a square is 36 minus b square or square in both side we have 16 is equal to 36 minus b square or b square is equal to 36 minus 16 that is 20. Also we have a square is equal to 6 square which is equal to 36 now this is the standard equation of an ellipse whose major axis is along the x axis therefore equation of the ellipse is given by x square upon a square that is 36 plus y square upon b square that is 20 is equal to 1 hence our answer is the equation of an ellipse whose vertices are plus minus 6 comma 0 and 4 guys plus minus 4 comma 0 is given by x square upon 36 plus y square upon 20 is equal to 1. So this completes the session hope you have understood it very well take care and have a good day.