Form the differential equation of the family of all the parabolas whose vertex is at origin and axis is y-axis.

Options

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

Answer

A

Explanation

Axis of parabola=Y axis
and vertex of parabola is (0,k)
`therefore` Equation of parabola is
`(x-0)^(2)=4a(y-k)`
`x^(2)=4ay-4ak`

On differentiate both sides w.r.t. 'x', we get
`2x=4a(dy)/(dx)`
`x=2a(dy)/(dx)`
`therefore(1)/(2a)=(1)/(x)(dy)/(dx)`
On differentiatingg both sides w.r.tt. 'x', we get
`(d)/(dx)((1)/(x)*(dy)/(dx))=(d)/(dx)((1)/(2a))`
`implies(1)/(x)*(d^(2)y)/(dx^(2))+(dy)/(dx)(-(1)/(x^(2)))=0`
`thereforex(d^(2)y)/(dx^(2))-(dy)/(dx)=0`