Let F be the family of ellipse whose centre is the origin and major axis is the y-axis. Then the differential equation of family F is

Options

A. ` (d^(2)y)/(dx^(2))+(dy)/(dx)(x(dy)/(dx)-y)=0`
B. ` xy(d^(2)y)/(dx^(2)) -(dy)/(dx) (x(dy)/(dx)-y)=0`
C. ` xy(d^(2)y)/(dx^(2))+(dy)/(dx)(x(dy)/(dx)-y)=0`
D. ` (d^(2)y)/(dx^(2))-(dy)/(dx)(x (dy)/(dx)-y)=0`

Answer

c

Explanation

Equation of family of ellipse is ` (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 `
On differentiating , we get
` rArr (2x)/(a^(2))+(2y)/(b^(2)) * (dy)/(dx) = 0`
` rArr x/(a^(2)) + y/(b^(2))* (dy)/(dx) = 0 `
Again , on differentiating , we get
` 1/(a^(2))+y/(b^(2))*(d^(2)y)/(dx^(2))+ ((dy)/(dx))^(2)1/(b^(2))=0`
` rArr (b^(2))/(a^(2)) + y ((d^(2)y)/dx^(2)) + ((dy)/dx)^(2) = 0 `
` rArr -y/x * (dy)/(dx) + y (d^(2)y)/(dx^(2)) + ((dy)/dx)^(2) = 0 `
` [ "from Eq. "(i) (b^(2))/(a^(2))= - y/x *(dy)/(dx)]`
` rArr xy (d^(2)y)/(dx^(2)) + (dy)/(dx) (x (dy)/(dx)-y )=0`