 Hello and welcome to this session. In this session we discuss the following question which says, find the center, the equations of the major and minor axis, the length and the equations of the lateral vector, focal and the direct tricks of the ellipse 5x square plus 28 plus 6 1 square plus 12 y minus 4 is equal to 0. The standard form for the equation of the ellipse is given as x square upon a square plus y square upon b square is equal to 1 where a is greater than b. For this equation the coordinates of the center is given as 0 0 then the coordinates of the focal are given as 0 then the equation of the major axis is given as y equal to 0. Equation of the minor axis is given as x equal to 0 the direct trusses x equal to plus minus a upon e length of the latest rectum then the equations of the two lateral vector which is the plural of the latest rectum are given as equal to plus minus t is given by e so this would be equal to square root of b square upon a square. Let's now proceed with the solution the given equation 4 is equal to 0 from these two terms we get 5 into x square plus taking 6 common from these two terms we have 6 into y square is equal to 0. Now the coefficients of x square and y square in the parenthesis are made unity. Now we will complete the squares so for this we have the whole here we have added to complete the square. Now this parenthesis is 20 so we would subtract 20 and as we have added 1 in this parenthesis so 6 into 1 is 6 so we will subtract this would be equal to 0. That is we now have 5 into the whole square plus 6 into y plus 1 the whole square minus 30 is equal to 0 plus 2 the whole square plus 6 into y plus 1 the whole square is equal to 30. Now dividing both sides by 30 we have 5 into upon 30 plus 6 into y plus 1 the whole square upon 30 is equal to 1 plus 30 so we now the whole square upon 6 is equal to 1. To get point minus x equal to x is equal to capital X minus y is equal to capital equation 1 y square upon 5 is equal to this equation of the inverse is given as 0 0 equal to 0 and capital Y equal to 0. Now this means that equal to 0 y plus 1 is equal to 0 i is equal to minus 1 would be the center of equation 1 equation 1 is given as minus 2 minus 1. The equation of the inverse is of this form when the equation of major axis is y equal to 0 so for the equation 2 this is given as capital Y equal to 0 this means plus 1 is equal to 0 or you can say that y is equal to minus the equation of the minor axis. Now the equation of this is given by x equal to 0 capital X equal to 0 which means that x plus 2 is equal to 0 or you can say that x is equal to minus 7 equation 1 of the latest rectum. Now solve the equation of the is equal to f is equal to 7 is given as 2 b square upon a these are b square and a here we have 2 into m root 3. This is the length of the latest rectum given by e and this is equal to square root of 1 minus upon a square e is equal to 1 minus of 1 upon is equal to 1 upon square root of 2. Now solve the latest rectum are given by x equal to e is equal to 1 upon root 6 so putting these values here 2 plus minus this is equal to plus minus 1 so x plus 2 is equal to plus minus 1 is equal to 1 2 is equal to minus 1 which means equal to minus 3 are the latest rectum equation 1. Equation 2 minus for a is root 6 and e is 1 upon root 6 comma 0 so this means the coordinates of the foci of equation 2 are plus minus 1 0 as we have equation 1 equal to plus minus 1 and capital Y equal to 0 s minus 1 2 is equal to 1 is equal to minus 1. This gives us x equal to minus 1 and x equal to minus 3 equal to 0 means equal to 0 which means that y is equal to minus 1 are given as minus 1 minus 1 2 plus minus a upon e is equal to plus minus which is 1 upon so this means that x plus 2 is equal to plus minus 6 equations of the direct this is equation 1. Equation 2 is equal to 4 and x equal to minus a plus 6 y square plus 12 equal to 0 is given as the latest rectum is equal to 5 root 2. Now we have the equations of the latest rectum are given as x equal to minus 1 x equal to minus 8 and x equal to 4. We were supposed to find out the solution of this question.