 Hi and welcome to the session I am Nipika here. Let's discuss a question which says form the equation of the family of circles in the first quadrant which touches the coordinate axis. Let's start the solution. Let DG node family of circles in the first quadrant catching the coordinate axis be the coordinates of the center of any member this family equation representing the family C is given by x minus a whole square plus y minus a whole square is equal to a square where a is the radius is any arbitrary constant. Let us give this equation as number 1. Now on differentiating both sides of equation 1 with respect to x we get 2 into x minus a plus 2 into y minus a into y dash is equal to 0 or x minus a plus y minus a into y dash is equal to 0 or this can be written as x plus y y dash minus a into 1 plus y dash is equal to 0 or x plus y y dash is equal to a into 1 plus y dash or a is equal to x plus y y dash over 1 plus y dash. Now on substituting the value of a in equation 1 we get x minus x plus y dash over 1 plus y dash whole square plus y minus x plus y y dash over 1 plus y dash whole square is equal to x plus y y dash over 1 plus y dash square or this can be written as x minus x over 1 plus y dash minus y y dash over 1 plus y dash whole square plus y minus y y dash over 1 plus y dash minus x over 1 plus y dash whole square equal to x plus y y dash over 1 plus y dash whole square this can be written as x into 1 plus y dash minus x over 1 plus y dash that is x y dash over 1 plus y dash minus y y dash over 1 plus y dash whole square plus y into 1 plus y dash minus y y dash over 1 plus y dash that is y over 1 plus y dash minus x over 1 plus y dash whole square and this is equal to x plus y y dash whole square over 1 plus y dash whole square or this can be written as let us take y dash square over 1 plus y dash whole square common from these two terms so we have y dash square over 1 plus y dash square into x minus y whole square plus now from these two terms let us take 1 over 1 plus y dash whole square common so we have this is equal to 1 over 1 plus y dash whole square into y minus x whole square and this is equal to x plus y dash whole square over 1 plus y dash whole square or this equation can be written as y dash square into x minus y whole square over 1 plus y dash whole square plus x minus y whole square over 1 plus y dash whole square and this is equal to x plus y y dash whole square over 1 plus y dash whole square or this can be written as y dash square into x minus y whole square plus x minus y whole square is equal to x plus y y dash whole square or this can be written as let us take x minus y whole square common from the left hand side we have x minus y whole square into y dash square plus 1 is equal to x plus y y dash whole square so this is a required differential equation hence the differential equation representing the given family of circles is x plus y y dash whole square is equal to x minus y whole square into 1 plus y dash square so this is the answer of your question I hope the solution is clear to you and you have enjoyed the session bye and take care