 Good morning friends, I am Purva and today I will help you with the following question. Form a differential equation representing the given family of curves by eliminating arbitrary constants a and b and we are given y square is equal to a into b square minus x square. Now as the order of a differential equation representing a family of curve is same as the number of arbitrary constants present in the equation corresponding to the family of curves. So here we shall differentiate the equation two times. So this is the key idea behind our question. Let us now begin with the solution. So here we are given y square is equal to a into b square minus x square. Let us mark this as equation one. Now as has been said in the key idea we shall differentiate equation one two times to eliminate the constants a and b. So differentiating equation one with respect to x we get. Now differentiating y square with respect to x we get 2y into dy by dx is equal to a into differentiating b square with respect to x we get 0 minus differentiating x square with respect to x we get 2x. Or we can write this as 2y into dy by dx is equal to minus 2a x. Now we cancel out 2 from both the sides and we get this implies y upon x into y dash because dy by dx can be written as y dash is equal to minus a. And we mark this as equation two. Now differentiating equation two with respect to x using quotient and product rule we get denominator that is x into differentiation of numerator. Now in numerator we have y into y dash so we apply product rule so we get y into y double dash that is first into differentiation of second plus second into differentiation of first. Now differentiation of y gives y dash minus numerator that is y into y dash into differentiation of denominator. So differentiation of x gives 1 upon denominator square that is x square is equal to now differentiating minus a with respect to x gives 0. And this implies x into y into y double dash plus y dash square minus y into y dash is equal to 0. And this further implies x into y into y double dash plus x into y dash square minus y into y dash is equal to 0. And this equation does not contain constants a and b so we write the above equation does not contain a and b. Hence the required differential equation is x into y into y double dash plus x into y dash whole square minus y into y dash is equal to 0. This is our answer. Hope you have understood the solution. Bye and take care.