 Hello and welcome to the session. In this session we are going to discuss some other forms of the ellipse. First of all we will discuss vertical ellipse. If in the given equation of an ellipse the coefficient of x square has a smaller denominator than the denominator of the coefficient of y square that is its major axis lies along y axis such an ellipse is called vertical ellipse that is if in the equation x square by a square plus y square by b square is equal to 1 b is greater than a then the major axis is of length to b is along the y axis and the length of the minor axis to a and it lies along the x axis in the given equation of the ellipse that is x square by a square plus y square by b square is equal to 1 where b is greater than a then the major axis is of length to b and lies along the y axis and the minor axis is of length to a and lies along the x axis. Now the centre lies at point O with the coordinates 00. O is the centre of the ellipse with the coordinates 00. Second its focal are given by f1 with the coordinates 0, b, e and f1 dash with the coordinates 0, minus b, e. There are two focal f1 and f1 dash with the coordinates 0, b, e and 0, minus b, e respectively. The directrix are given by the equations y is equal to plus minus b upon e. Now the directrix z and z dash are given by the equations y is equal to b upon e and y is equal to minus b upon e where e is the eccentricity given by the equation a square is equal to b square into 1 minus e square. We can write e is equal to square root of 1 minus a square upon b square. Its vertices are given by with the coordinates 0, b and a dash with the coordinates 0, minus b. Its vertices are given by the point a with the coordinates 0, b and the point a dash with the coordinates 0, minus b. Thus we know that if the denominator of y square upon b square is greater than the denominator of x square by a square, that is b square is greater than a square then it is a vertical ellipse. This is the required vertical ellipse with center o with coordinates 0, 0, vertices a and a dash with coordinates 0, b and 0, minus b respectively, focal f1 and f1 dash having coordinates 0, b, e and 0, minus b, e respectively. Now we will discuss axis parallel to the axis of coordinates. Now the coordinate axis are x axis and y axis. Let the axis parallel to the coordinate axis be denoted by x dash and y dash. This is an ellipse with center o dash having coordinates h, k. Now let p be any point on the ellipse having the coordinates x, y with respect to some o, y the axis of coordinates and x dash, y dash are the coordinates with respect to x dash and o dash, y dash. Therefore the equation of the ellipse with respect to dash x dash and o dash, y dash are the coordinate axis x dash square by a square plus y dash square by b square is equal to 1. As we know that x is equal to x dash plus h and y is equal to y dash plus k. This implies that x dash is equal to x minus h and y dash is equal to y minus k. Therefore the above equation will become x minus h the whole square by a square plus y minus k the whole square by b square is equal to 1 which is the required equation of the ellipse. Therefore the equation of the ellipse with center having coordinates h, k and the axis of the coordinates is of the form minus h the whole square by a square plus y minus k the whole square by b square is equal to 1. On simplification the above equation will reduce to the form lx square plus my square plus 2gx plus 2fy plus c is equal to 0 where lx are two different positive numbers. Therefore we can also conclude that the equation of the form x square plus my square plus 2gx plus 2fy plus c is equal to 0 represents an ellipse whose axis are the axis of the coordinates. Now we shall discuss the general equation of the ellipse. We can find the equation of the ellipse when the coordinates of the focus eccentricity and equation of the corresponding directrix are given. From the definition of ellipse we know that the distance of any point p on the ellipse on the focus is equal to e into the distance of p from the directrix. Let the focus f be given by the coordinates alpha, beta eccentricity, b and the equation of the corresponding directrix be given by lx plus m y plus m is equal to 0. Let p with the coordinates x, y be any point on the ellipse then by using the definition of the ellipse we can say that square root of m is alpha d whole square plus y minus beta d whole square is equal to e into modulus of n y plus n upon square root of n square plus n square where the value of n square plus n square cannot be 0. On squaring both sides we get n square plus n square into x minus alpha d whole square plus y minus beta d whole square is equal to e square into n x plus n y plus n d whole square and for the simplification it will reduce to the form of 2 h x y plus b y square plus 2 g x plus 2 f y plus c is equal to 0. This is the general equation of the ellipse from the above equation we can say that the equation of an ellipse certain degree equation minus ab is less than 0 implies that x square is less than ab where h b are the constants in a certain degree equation is less than ab then it represents an ellipse this completes our session hope you enjoyed this session.