 Hello and welcome to the session. In this session we will discuss how to simplify an equation of a conic using translation of axis. A conic can be a parabola, an ellipse, a hyperbola or a circle. Now we are going to discuss translation of axis. A translation of coordinate axis occurs when the new coordinate axis have the same direction as the original coordinate axis and are also parallel to the original coordinate axis. Now let us draw the coordinate axis which are the original coordinate axis and p is the point with coordinates x, y in the original system. Now let us translate the origin O, h units horizontally and k units vertically to point O'. Now we draw the new axis at new origin O' by drawing vertical and horizontal lines. These are the translated axis, x' axis and y' axis and the new origin O'. With respect to this translated axis the point p will now have coordinates x', y', if the coordinates of the original O' of the translated system are h, k related to the original system then the new and the old coordinates are related using the following result 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 axis. Here we should note that this is the original axis with x and y axis and this is the translated axis with x' and y' axis. We should note that using this result we can also find the coordinates of any point say p in original as well as in new coordinate axis. Coordinates of point p in new axis are given by point p with coordinates x' y', that will be equal to point p with coordinates x minus h, y minus k. Similarly original coordinates of point p are given by point p with the coordinates x, y and it is given by point p with the coordinates x' plus h, y' plus k. Now we know that the general equation of any conic is given by ax square plus bxy plus cy square plus dx plus ey plus f is equal to 0 where both a and c cannot be 0. Also when we translate the axis then the xy term will be missing in the equation because axes are parallel to the original axis only. Thus given any equation of a conic we can find the equation of the conic in the translated axis. Let us see the following example transform the equation x square minus y plus 6x plus 10 is equal to 0 when the axes are translated to the new origin with coordinates minus 3, 1. That is now we will find the equation in x' y' axis. That is in this question we have to rewrite the given equation replacing all coordinates that is x and y variables with the new coordinates that is x' y' variables. So here the coordinates hk are given as minus 3, 1. So h is equal to minus 3 and k is equal to 1. Now to transform in x' y' axis we use the result that is x is equal to x' plus h and y is equal to y' plus k. So now we put the values of h and k in these equations and we get x is equal to x' plus h and h is minus 3 so it is equal to x' minus 3 and y is equal to y' plus k and k is equal to 1 so we have y is equal to y' plus 1. Now putting the values of x is equal to x' minus 3 and y is equal to y' plus 1 in the given equation we get The given equation is x square minus y plus 6x plus 10 is equal to 0 and when we put the values of x and y in this equation we get x' minus 3 whole square minus y' plus 1 the whole plus 6 into x' minus 3 the whole plus 10 is equal to 0 which implies that Now x' minus 3 whole square can be written as x' square plus 9 minus 2 into x' into 3 by using the formula of a minus b whole square that is equal to a square plus b square minus 2ab minus y' minus 1 plus 6 into x' that is 6x plus 6 into minus 3 that is minus 18 plus 10 is equal to 0 Now this further implies that x' square plus 9 minus 6x prime minus y prime minus 1 plus 6x prime minus 18 plus 10 is minus 8 is equal to 0 Now we cancel like terms of 6x prime and this implies that x' square minus y prime plus 9 minus 1 minus 8 is equal to 0 This further implies that x' square minus y prime plus 9 now minus 1 minus 8 is minus 9 is equal to 0 which implies that x' square minus y prime is equal to 0 which implies that x' square is equal to y prime all we can write it as y prime is equal to x' square so the transformed equation in the translated axis will be y prime is equal to x' square and this is the equation of the parabola Thus in this session we have discussed how to simplify an equation of a conic using translation of axis This completes our session hope you enjoyed this session