 Hello and welcome to this session. In this session we will discuss about the conic section ellipse. Basically an ellipse is the set of all points in a plane the sum of whose distance is from two fixed points in the plane is a constant. This is an ellipse consider two fixed points f1 and f2 these are the two focuses of the ellipse or you can say foci which is the plural of focus then the midpoint of the line segment joining the foci that is f1 and f2 that is this point is called the center of the ellipse that is this point O is the center of the ellipse the line segment through the foci of the ellipse is called the major axis that is this line segment is the major axis let this be point A and this be point B so we have the line segment AB is the major axis and the line segment through the center and perpendicular to the major axis is the minor axis that is this line segment is the minor axis consider this point to be C and this B so we have the line segment CD is the minor axis the end points of the major axis that is the points A and B they are the vertices of the ellipse we have length of the major axis of the ellipse is equal to 2A and length of the minor axis of the ellipse is 2B and the distance between the two foci that is f1 and f2 is 2C that is this distance is 2C length of semi major axis is A and length of semi minor axis is B now next we have relationship between any major axis semi minor axis the distance of the focus from the center of the ellipse is given by A squared equal to B squared plus C squared or we can write it as C is equal to square root A squared minus B squared where C is the distance of the focus from the center A is the length of the semi major axis and B is the length of semi minor axis now we discuss special cases of an ellipse now from this equation A squared equal to B squared plus C squared we have C squared equal to A squared minus B squared now in this equation we keep A fixed and we vary C from 0 to A and the resulting ellipses will vary in shape like in the first case when we take C equal to 0 then both the foci merge together with the center of the ellipse and we have A squared is equal to B squared that is A is equal to B and thus the ellipse becomes circle so we say that circle is a special case of an ellipse next when we take C is equal to A then we get B is equal to 0 and thus the ellipse reduces to the line segment f and f2 that is the line segment joining the two foci next we have eccentricity eccentricity is the ratio of the distances from the center of the ellipse to one of the foci one of the vertices of the ellipse it is denoted by E so we have E is equal to C upon A from here we get C is equal to AE that is focuses at a distance of AE from the center next we discuss standard equations of an ellipse we have f1 and f2 are the foci of the ellipse and O is the center of the ellipse we take the coordinates of the focus f1 of the ellipse as minus C0 and coordinates of the focus f2 of the ellipse as C0 this is the origin O with coordinate 00 now A is the vertex of the ellipse it has coordinates minus A0 and the vertex B has coordinates A0 then C has coordinates 0 B and B has coordinates 0 minus B so we have equation of an ellipse with center as the origin and major axis along the x-axis is given by x square upon a square plus y square upon v square equal to 1 now from here we have that the ellipse lies between the lines x equal to minus a and x equal to a and touches these lines similarly we have the ellipse lies between the lines y equal to minus B and y equal to B and touches these lines we have another standard equation of the ellipse that is x square upon b square plus y square upon a square equal to 1 now these two are the standard equations of the ellipse from these two standard equations we have some observations like we can say that ellipse is symmetric with respect to both the coordinate axis then foci always lie on the major axis and the major axis can be determined by finding the intercepts on the axis of symmetry that is if we have the coefficient of x square has larger denominator then the major axis is along the x-axis and if we have coefficient of y square has larger denominator then the major axis is along y-axis next is the latest rectum latest rectum of an ellipse is the line segment perpendicular to the major axis through any of the foci and whose endpoints lie on the ellipse like this line segments PQ and EF are the latest rectum of the ellipse now length of the latest rectum of an ellipse is given by 2B square upon a let's consider the equation of the ellipse given by x square upon 16 plus y square upon 9 equal to 1 now this is of the form x square upon a square plus y square upon b square equal to 1 so when we compare these two we get that a square is equal to 16 and b square is equal to 9 that is we have a is equal to 4 and b is equal to 3 now let's find out C C is equal to square root a square minus b square that is square root 16 minus 9 so we get C is equal to square root 7 then length of the major axis is given by 2A that is 8 then length of the minor axis is given by 2B that is equal to 6 now let's find out the coordinates of the vertices let one vertex be given by a with coordinates minus a 0 so it would be minus 4 0 and another vertex B with coordinates a 0 that is 4 0 so these are the two vertices of the given ellipse now let's find out the coordinates of the two foci let one focus be F1 with coordinates minus C 0 that is minus square root 7 0 and the other focus be F2 with coordinates C 0 that is square root 7 0 eccentricity E of the ellipse is given by C that is square root 7 upon A that is 4 then we have length of the latest rectum is given by 2B square that is 2 multiplied by 9 upon A that is 4 so we get this is equal to 9 upon 2 9 upon 2 is the length of the latest rectum this completes the session hope you have understood the conic section ellipse