 Good morning friends, I am Pudva and today I will help you with the following questions from the differential equation of the family of circles having center on y-axis and radius 3 units Let us now begin with the solution Now, let's see with the family of circles having center on y-axis radius equal to 3 units and And let the center of the member of the family C be 0 comma a So this is the family of circles C having center on y-axis and the center has coordinates 0 comma a Then we have the equation of the circle is x square plus y minus a whole square is equal to 3 square and We can write this as this implies x square plus now y minus a whole square can be written as y square plus a square minus 2 a y and This is equal to 9 because 3 square is equal to 9 Let us mark this as equation 1 since we have to eliminate a so Differentiating equation 1 with respect to x we get Differentiating x square gives 2x plus Differentiating y square gives 2y into y dash Differentiating a square gives 0 so we have plus 0 minus Differentiating 2 a y gives 2 a y dash is equal to now Differentiating 9 gives 0 Now we can write this as this implies x plus y into y dash is equal to a into y dash And this further implies x plus y into y dash upon y dash is equal to a We mark this as equation 2 Now substituting the value of a n x square plus y minus a whole square is equal to 9 we get x square plus Y minus now a is equal to x plus y into y dash upon y dash Whole square is equal to 9 this implies x square plus Now taking the equation we get y into y dash minus x minus y into y dash upon y dash Whole square is equal to 9 now we can cancel out y into y dash We get this implies x square plus minus x upon y dash whole square is equal to 9 We can write this as this implies x square plus x square upon y dash square is equal to 9 And this further implies x square into y dash square Plus x square is equal to 9 into y dash square This implies x square into y dash square plus x square minus 9 y dash square is equal to 0 And we can write this as this implies Now taking out y dash square common from these four terms we get x square minus 9 Into y dash square plus x square is equal to 0 Now this equation does not contain any constant hence. This is the required differential equation So we write the above equation Does not contain any constant Hence This is the required differential equation Therefore the required answer is x square minus 9 Into y dash square plus x square is equal to 0 Hope you have understood the solution buy and take care