 Hello and welcome to the session. In this session we are going to discuss the following question and the question says that find centered and radius of circle given by the equation x square plus 5x plus y square minus 6y plus 25 by 2 is equal to 0 by method of completing square. We know that standard form of equation of circle with center hk and radius r is given by x minus h whole square plus y minus k whole square is equal to r square. With this key idea we shall proceed to the solution given equation of the circle is x square plus 5x plus y square minus 6y plus 25 by 2 is equal to 0 and we have to find its center and radius. So we will first convert it in standard form for that we need to make perfect squares. For this first we keep terms with variable x and variable y on one side of the equation and bring the constant term on the other side. So the equation becomes x square plus 5x plus y square minus 6y is equal to minus of 25 by 2. Now we put the x variable and y variable terms in separate brackets so we get x square plus 5x the whole plus y square minus 6y the whole is equal to minus of 25 by 2 and now we will make perfect squares in the brackets. Now for x variable terms we divide the coefficient of x by 2 and then square the result we obtain and then we add this obtained number inside the bracket and also add this number to the right hand side of the equation so that we don't change the original equation. Here we have x square plus 5x so we will add square of 1 by 2 of coefficient of x that is we multiply the coefficient of x by 1 by 2 and then we square it and this is equal to 25 by 4 and we will follow the same method for terms in y variable here we have y square minus 6y so we will add square of 1 by 2 of coefficient of y that is we have 1 by 2 into minus 6 the whole square which is equal to minus 3 whole square that is equal to 9 so we add 25 by 4 and 9 in x and y terms brackets respectively also we will add 25 by 4 and 9 to right hand side of the equation so we have x square plus 5x plus 25 by 4 the whole plus y square minus 6y plus 9 the whole is equal to minus of 25 by 2 plus 25 by 4 plus 9 which implies that x square plus 5x plus 25 by 4 the whole plus y square minus 6y plus 9 the whole is equal to now taking the LCM here we have 4 in the denominator and in the numerator we have minus 50 plus 25 plus 36 which further implies that x square plus 5x plus 25 by 4 the whole plus y square minus 6y plus 9 the whole is equal to now minus 50 plus 25 is minus 25 plus 36 will be equal to 11 upon 4 now this can be written as x square plus 5x plus 5 by 2 whole square the whole plus y square minus 6y plus 3 square the whole is equal to 11 by 4 and we know that a minus b whole square is equal to a square minus 2ab plus b square and a plus b whole square is equal to a square plus 2ab plus b square so using these formulae we have x plus 5 by 2 whole square plus y minus 3 whole square is equal to 11 by 4 which is the standard form of the equation of the circle that is x minus h whole square plus y minus k whole square is equal to r square now on comparing these two equations we have h is equal to minus of 5 by 2 k is equal to 3 and r square is equal to 11 by 4 so we say that its center is given by the ordered pair minus of 5 by 2 3 and we have r square is equal to 11 by 4 which implies that r is equal to square root of 11 by 4 that is by taking positive square root and it implies that r is equal to square root of 11 by 2 so the given circle has center which coordinates minus 5 by 2 3 and radius is equal to square root of 11 by 2 this is the required answer this completes our session hope you enjoyed this session