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Consider the family of all circles whose...

Consider the family of all circles whose centers lie on the straight line `y=x` . If this family of circles is represented by the differential equation `P y^+Q y^(prime)+1=0,` where `P ,Q` are functions of `x , y` and `y^(prime)(h e r ey^(prime)=(dy)/(dx),y^=(d^2y)/(dx^2)),` then which of the following statements is (are) true? (a) `( b ) (c) P=y+x (d)` (e) (b) `( f ) (g) P=y-x (h)` (i) (c) `( d ) (e) P+Q=1-x+y+y +( f ) (g)(( h ) (i) y^(( j )prime( k ))( l ) ( m ))^(( n )2( o ))( p ) (q)` (r) (s) `( t ) (u) P-Q=x+y-y -( v ) (w)(( x ) (y) y^(( z )prime( a a ))( b b ) ( c c ))^(( d d )2( e e ))( f f ) (gg)` (hh)

A

(a) `P=y+x`

B

(b) `P=y-x`

C

(c) `P+Q=1-x+y+y'+(y')^(2)`

D

(d) `P-Q=x+y-y'-(y)^(2)`

Text Solution

Verified by Experts

The correct Answer is:
(b,c)
Since, centre lies on Y = x.
`therefore` Equation of circle is
`x^(2) +y^(2)-2ax-2ay+c=0`
On differentiating, we get
`2x=2yy'-2a-2ay'=0`
`rArr x+yy'-a-ay'=0`
`rArr a=(x+yy')/(1+y')`
Again differentiating, we get
`0=((1+y')[1+yy''+(y')^(2)]-(x+yy') cdot (y''))/((1+y')^(2))`
`rArr (1-y')[1+(y')^(2) +yy'']- (x+yy')(y'')=0`
`rArr 1+y'[(y')^(2)+y'+1]+y''(y-x)=0`
On comparing with `Py''+Qy'+1=0,` we get
`P =y - x`
and `Q = (y')^(2) +y'+1`
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