Question

Find the differential equation of all the circles which pass through the origin and whose centres lie on y-axis.
A. `(x^(2)-y^(2))(dy)/(dx)-2xy=0`
B. `(x^(2)-y^(2))(dy)/(dx)+2xy=0`
C. `(x^(2)-y^(2))(dy)/(dx)-xy=0`
D. `(x^(2)-y^(2))(dy)/(dx)+xy=0`

Answer 1

Correct Answer - A
The equation of the family of circles passing through the origin and having centres on y-axis is
`x^(2)+(y-a)^(2)=a^(2)rArrx^(2)+y^(2)-2ay=0" ...(i)"`
Differentiating w.r.t. x, we get
`2x+2y(dy)/(dx)-2a(dy)/(dx)=0rArra=(x+y(dy)/(dx))/((dy)/(dx))=x(dx)/(dy)+y`
Substituting the value of a in (i), we obtain
`x^(2)+y^(2)-2y(x(dx)/(dy)+y)=0`
`rArr" "x^(2)-y^(2)-2xy(dx)/(dy)=0rArr(x^(2)-y^(2))(dy)/(dx)-2xy=0`