 Good morning friends, I am Purva and today I will help you with the following question form the differential equation of the family of hyperbolas having foci on x-axis and center at origin. Let us now begin with the solution. Now since we are required to find the differential equation of the family of hyperbolas having foci on x-axis and center at origin so first we should find the equation of one member of the family H of the hyperbolas. Now equation of one member of family H is h square by a square minus y square by b square is equal to 1 and we mark this as equation 1 and here a and b are constants. Now since this equation contains two arbitrary constants a and b so in order to eliminate them we shall differentiate equation 1 two times so differentiating 1 with respect to x we get differentiating x square by a square we get 2x upon a square minus differentiating y square by b square we get 2y into dy by dx upon b square is equal to differentiating 1 we get 0 or we can write this as minus 2y into now dy by dx is equal to y dash so we have y dash upon b square is equal to minus 2x upon a square and this implies y into y dash upon x is equal to b square upon a square and we mark this as equation 2. Now differentiating equation 2 with respect to x we get now we apply quotient and product rule on left hand side so we get denominator that is x into differentiation of numerator now in numerator we have y into y dash so we apply product rule here and we get first that is y into differentiation of second now differentiation of y dash gives y double dash plus second that is y dash into differentiation of first now differentiation of y gives y dash minus numerator that is y into y dash into differentiation of denominator now differentiation of x gives 1 so we write into 1 upon denominator square that is x square is equal to now differentiation of b square by a square gives 0 this implies now opening the bracket we get x into y into y double dash plus x into now y dash into y dash gives y dash square minus y into y dash into 1 gives y into y dash and this is equal to now x square into 0 gives 0 and this is the required differential equation as it is independent of the constants a and b hence we write our answer as x into y into y double dash minus x into y dash square minus y into y dash is equal to 0 this is our answer hope you have understood the solution bye and take care