 Hello and welcome to the session. In this session we will discuss rectangular hyperbola and asymptotes. To find, let us discuss rectangular hyperbola. Now, we know that the standard equation of the hyperbola is x square over a square minus y square over b square is equal to 1. Here are the constants and here the length of the transverse axis of this parabola is equal to 2f and the length of the conjugate axis is equal to 2b. Now, if a i is equal to b, that is if the length of the transverse conjugate axis, the hyperbola, the rectangular hyperbola and its equation is, that is by putting b is equal to a in the equation of this hyperbola, we get x square over a square minus y square over a square is equal to 1. This implies x square minus y square is equal to a square. Now, let us discuss the eccentricity of the rectangular hyperbola. Now, we know that if e is the eccentricity of the hyperbola, x square over a square minus y square over b square is equal to 1. Then, we will be equal to 1 plus b square over a square and this implies equal to a square into e square minus 1 the whole. And we also know that the eccentricity of the hyperbola is greater than 1. Now, b square is equal to a square into e square minus 1 the whole. For the rectangular hyperbola, a is equal to b. For rectangular hyperbola, therefore replacing b with a in this equation, it will be a square is equal to a square into e square minus 1 the whole. Which implies a square over a square is equal to e square minus 1 which further implies 1 is equal to e square minus 1 and this implies 1 plus 1 which is true. And this gives e is equal to the hyperbola e is equal to minus root 2 the hyperbola is equal to equations of rectangular hyperbola. Now, if another variable x is equal to equal to t of t, then these will be here the third way that is t is the variable parameter. Now, the equation is equal to a square which is the equation of rectangular hyperbola. Then, we transform f is equal to a square plus b square therefore equal to b in case of rectangular hyperbola. So, this is equal to 2 a square therefore which is equal to a square by 2. A that is this equation into c by t will give two values that is x is equal to c t and y is equal to c by t. We always have the equation again so that the point c by t lies to c by t. Now, an asymptote to your curve is the tangent to your curve contact is at infinity. Or in particular the curve to a point v on the hyperbola v on the curve moves farther and farther away from the origin. It may happen that the distance between the point v and these fixed levels on this side this fixed line is called also the asymptote of the given hyperbola. That means these fixed line will touch the curve that is the hyperbola and now let y is equal to mx plus c is the asymptote. The given hyperbola over a square minus y square over b square is equal to 1. That is here we will see how to derive the equation of the asymptote of the given hyperbola. Now, let this be equation a so putting y is equal to mx plus c in the equation a we get over a square minus mx plus c whole square over b square is equal to 1. Which implies that is on solving this figure 1 over a square minus m square over b square the whole into x square minus 2 mc over b square into x minus c square over b square plus 1 the whole is equal to 0. Now the line y is equal to mx plus c that is the asymptote that is the hyperbola which is given by equation a at infinity that is the roots the equation which is the equation b are infinite. This means the coefficient of x square and the coefficient of x in this equation is equal to 0. So putting the coefficients equal to 0 this implies plus minus b over a and c is equal to 0. Now putting the values of m y is equal to mx plus c we get y is equal to plus minus b over a into x which is 0 which implies y is equal to plus minus b over a into x. And these are the equations of the asymptotes of the given hyperbola. So diagrammatically we can see that these are the equations of the asymptotes of the given hyperbola whose equation is x square over a square minus y square over b square is equal to 1. So in this session we have learnt about rectangular hyperbola and asymptotes. And this concludes our session hope you all have enjoyed the session.