 Hello and welcome to the session. In this session we will discuss equation which says that for the parabola 2y square minus 6x plus 6y plus 33 is equal to 0. Find first part segments of vertex and focus, second part length of latest vector, third part equation of axis and fourth part equation of directorates. Now before starting the solution of this question we should know some results. And that is the summary of some facts about four parabolas in standard form. First of all let us discuss the coordinates of vertex for these four parabolas. Now for the parabola of the form y square is equal to 4ax the coordinates of vertex are 00. For this parabola that is y square is equal to minus 4ax the coordinates of vertex are 00. And for these two also the coordinates of vertex are 00. Then the coordinates of focus for this first form of parabola is 0 for the second form is a0 for the third form is 0 in and for the fourth form is 0 minus a. Now the length of the latest vector for the parabola y square is equal to 4ax is 4a. For the second form of parabola it is 4a. For the next two forms it is also 4a and 4a. Now let us discuss the equation of x parabola y square is equal to 4ax is equal to 0. For the parabola y square the parabola x square is equal to 4a is equal to 0. And for this parabola x is equal to 0. The parabola it is x is equal to a y is equal to minus a and for the fourth form of parabola it is y is equal to a. Now these results will work out as a t idea and we will start with the solution. Now the equation of the equation of parabola y square minus 6x plus 6y plus 33 is equal to 0. 6y is equal to 6x minus 33. Now dividing throughout y2 this implies y square plus 3y is equal to 3x minus 33 by 2. The equation that is this equation left inside we will add and subtract. So this will be y square plus 3y till the y is 3 by 2 whole setting. The square y square plus 3y is equal to 3x minus 33 by 2 this implies y plus 3 by 2 whole square is equal to 3x minus 33 by 2 plus 9 by 4. This further implies y plus 3 by 2 whole square is equal to 3x and on solving this it will give minus 57 by 4 implies y plus 3 by 2 whole square is equal to now from this taking 3 common it will be 3 into x minus 19 by 4 the whole. In the standard form which is not in the standard form of the equation of parabola then we will shift to minus plus beta by putting capital X minus alpha and capital Y is equal to y plus minus beta. That is if beta and put in here then we will shift the origin to plus alpha plus beta by putting capital X is equal to minus beta and if beta then we will shift plus beta by putting capital X is equal to x plus alpha is equal to y plus beta. So this in the form y this beta whole square is equal to 3 into x minus alpha the whole. Now to express this equation in the standard form of parabola the signs of alpha and beta by putting is equal to y is equal to y by the origin which is minus 3 by minus 19 therefore y is equal to y plus 3 by 2. Now this equation number 1 then we get y square is equal to n is equal to now this can be written as 4 into 3 by 4 into x. Now this in the form y square is equal to 4x the equation square is equal to 4x now we are into 3 by 4 is equal to 3 by 4. That is y square is equal to 4 into 3 by 4 into x which is of the form y square is equal to 4x which is given the key idea. Now capital X is equal to x minus 19 by 4 and capital Y is equal to y plus 3 by 2. 3 to 0 implies x minus 19 by 4 is equal to 0 which implies x is equal to 19 by 4 and capital Y is equal to 0 implies 2 is equal to 0 which further implies y is equal to minus 3 by 2. Therefore x of vertex is 3 by 2. Now x to new x is 0. Therefore y is equal to 0 is equal to 3 by 4 and y is equal to 3 by 4. Now 3 by 4 implies now 19 by 4 to 3 by 4 which implies x is equal to 19 by 4 plus 3 by 4 will give 22 by 4. This implies x which will be y plus 3 by 2 is equal to 0 which implies y is equal to minus 3 by 2. Therefore of 4 plus 3 by 2. Now in the second part we have to find the length of latest one by using this result which is given in the key idea. Length of the latest vector will be equal to 4a 3 by 4. So this will be 4 into 3 by 4 which is equal to 3. Now in the next part the equation of now using this result which is given in the key idea is equal to y plus 3 by 2 is equal to 0 which implies y is equal to minus 3 by 2. Now in the next part we have to find the equation of direct rates. Now using this result which is given in the key idea capital X is equal to minus a. That is capital X is equal to now a is 3 by 4 so it will be minus 3 by 4. 3 by 4 implies now capital X is equal to 19 by 4 will be equal to minus 3 by 4 minus 3 by 4 which gives x is equal to 16 by 4 which is equal to 4. Therefore of the given question and that's all for this session. Hope you all have enjoyed the session.