 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that transform the equation x square plus y square plus 4x minus 6y minus 51 is equal to 0 when the axes are translated to the new origin which coordinates minus 2, 3 also draw its curve in the new axes. Before starting the solution of this question, we should know our result. If the coordinates of the origin O' of the translated system are hk related to the original system then the new and the old coordinates are related using the following result and that is x is equal to x' plus h and y is equal to y' plus k. Using this result, we can obtain the equation of the conic in the original as well as translated axes. With this key idea we shall move on to the solution. In this question we have to transform the equation that is x square plus y square plus 4x minus 6y minus 51 is equal to 0 when the axes are translated to the new origin which coordinates minus 2, 3. Now see that here we are given the equation in x and y variables that is curve is in the original coordinate axes with origin at 0, 0 and we have to find the transformed equation when the axes are translated to minus 2, 3 that is now we will find the equation in x' y' axes that is in this question we have to rewrite the given equation replacing old coordinates that is x and y variables with the new coordinates that is x' y' variables. So here the ordered pair hk will be given by the ordered pair minus 2, 3 which implies that h is equal to minus 2 and k is equal to 3. Now to transform it in x' y' axes we use the result given in the key idea that is x is equal to x' plus h and y is equal to y' plus k. Now we put the values of h and k in these equations and we get x is equal to x' plus or minus 2 that is equal to x' minus 2 and y is equal to y' plus 3. Now we put these values of x and y in the given equation we get x square that is x' minus 2 whole square plus y square that is y' plus 3 whole square plus 4 into x that is 4 into x' minus 2 the whole minus 6 into y that is 6 into y prime plus 3 the whole minus 51 is equal to 0. Now we simplify this equation and we get now here we have x' minus 2 whole square here we will apply the formula that is a minus b whole square is equal to a square minus 2ab plus b square so here we have x' square minus 4x prime plus 4 plus now here we have y prime plus 3 whole square and here we will apply the formula a plus b whole square which is equal to a square plus 2ab plus b square so here we will write y prime square plus 6y prime plus 9 plus 4 into x prime that is 4x prime plus 4 into minus 2 that is minus 8 now minus 6 into y prime will be minus 6y prime minus 6 into 3 will be minus 18 minus 51 is equal to 0. Now combining the like terms here we get x prime square plus y prime square plus 4x prime minus 4x prime the whole plus 6y prime minus 6y prime the whole plus 4 plus 9 minus 8 minus 18 minus 51 is equal to 0 now we cancel the terms and here we get x prime square plus y prime square now solving this we get minus 64 is equal to 0 this implies that x prime square plus y prime square is equal to 64 and this is the equation of the circle with radius 8 as we can write this equation as x prime square plus y prime square is equal to 8 square now we will draw the curve of the circle in new axis first we draw the original coordinate axis with origin O now new origin is given to be the ordered pair minus 2 3 so we translate the origin 2 units left and 3 units up at this point we draw the new coordinate axis O prime X prime and O prime Y prime here we name this new origin as O prime with coordinates minus 2 3 now this is the required curve of the equation x prime square plus y prime square is equal to 64 thus we have transformed the given equation into this equation when the axis are translated to the new origin with coordinates minus 2 3 and this is the required curve in the new axis this completes our session hope you enjoyed this session