 Hello and welcome to the session. In this session we will discuss a question which says that show that the circle x plus y square plus 4 into x plus y the whole plus 4 is equal to 0. That shows the coordinate axis. Find the equation of the circle which passes through the points of intersection of this circle with the line x plus y plus 2 is equal to 0 and has its center at the origin. Now before starting the solution of this question we should know what is that. And that is let x is given by x square plus y square plus 2 gx plus 2 fy plus c is equal to 0 and p is given by mx plus ny plus m is equal to 0 below equations of a circle and a line respectively. Then plus lambda p is equal to 0 that is p is this so it will be plus y square plus 2 at y plus c plus lambda into a x plus and y plus m the whole is equal to 0 circle and this line. So it will be through the intersection of 1 we will work out as a key idea for solving out this question with the solution. Now the equation of the circle is given to us and we have to show that it touches the coordinate axis. Now given x plus y the whole let us name it as 1. Now for the points of intersection circle plus 0 plus 4 into x plus 0 the whole plus 4 is equal to 0 which implies x square plus 4x plus 4 is equal to 0. Now this is a quadratic equation in x so for solving this we will split the middle term and it will be x square plus 2 plus 2x plus 4 is equal to 0 which further implies into x plus 2 the whole is equal to 0 which further implies the whole into x plus 2 the whole is equal to 0. Now here putting u of the factor equal to 0 this implies minus 2 the circle the intersection of this circle that is this circle with the y axis for the points of intersection circle which is given by equation number 1 with the y axis equal to 0 in equation number 1 we get it plus 4 is equal to 0 which further implies on splitting the middle term y square plus 2y plus 2y plus 4 is equal to 0 which implies y plus 2 the whole plus 2 into y plus 2 the whole is equal to 0 which further implies y plus 2 the whole into y plus 2 the whole is equal to 0 by putting u of the factor equal to 0 it will be y is equal to minus 2 and minus 2 that is 0 minus 2 the equation of the circle which passes through the points of intersection of this circle with this line and have it centered at the origin now given u of the line plus 2 is equal to 0 now let us know it is in this result which is given by key idea the equation the section which is given by equation number 1 that is this circle the line which is given by equation number 2 plus y square plus 4y plus 4 plus lambda into is equal to 0 plus y square plus lambda the whole into x lambda the whole into y plus 4 plus 2 lambda is equal to 0 now let us see this equation number 3 now we know that the general equation of the circle is this and here the permanents of center are minus g and minus f of this circle by 2 and minus of 4 plus lambda the whole by 2 it is given that the center that is to have this circle the center of the circle which is given by equation number 3 and this is also the center of the same circle thus x coordinate plus lambda by 2 is equal to equal to 0 which further implies lambda is equal to minus 4 now this is the equation number 3 lambda is equal to minus 4 in equation number 3 pull the whole into x plus 4 minus 4 the whole into y plus 4 minus 8 is equal to 0 4 is equal to 0 which further implies plus y square is equal to 4 this is the required equation of this question so that's all for this session hope you all have enjoyed the session