The differential equation of all parabolas whose axis are parallel to the y-axis is (a) `( b ) (c) (d)(( e ) (f) d^(( g )3( h ))( i ) y)/( j )(( k ) d (l) x^(( m )3( n ))( o ))( p ) (q)=0( r )` (s) (b) `( t ) (u) (v)(( w ) (x) d^(( y )2( z ))( a a ) x)/( b b )(( c c ) d (dd) y^(( e e )2( f f ))( g g ))( h h ) (ii)=C (jj)` (kk) (c) `( d ) (e) (f)(( g ) (h) d^(( i )3( j ))( k ) y)/( l )(( m ) d (n) x^(( o )3( p ))( q ))( r ) (s)+( t )(( u ) (v) d^(( w )2( x ))( y ) x)/( z )(( a a ) d (bb) y^(( c c )2( d d ))( e e ))( f f ) (gg)=0( h h )` (ii) (d) `( j j ) (kk) (ll)(( m m ) (nn) d^(( o o )2( p p ))( q q ) y)/( r r )(( s s ) d (tt) x^(( u u )2( v v ))( w w ))( x x ) (yy)+2( z z )(( a a a ) dy)/( b b b )(( c c c ) dx)( d d d ) (eee)=C (fff)` (ggg)

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)`.
`:.` Equation of parabola is
`(x-0)^(2) = 4a (y-k)`
`x^(2) = 4ay - 4ak`

On differentiate both sides w.r.t. both sides w.r.t. 'x' we, get
`2x= 4a (dy)/(dx)`
`x =2a '(dy)/(dx)`
`:. 1/(2a) = 1/x (dy)/(dx)`
On differentiating both sides w.r.t., 'x' we get
`d/(dx) (1/x.(dy)/(dx)) = (d)/(dx) (1/(2a))`
`rArr (1)/(x).(d^(2)y)/(dx^(2)) +(dy)/(dx) (-(1)/(x^(2))) = 0`
`:. x'(d^(2)y)/(dx^(2)) - (dy)/(dx)= 0`