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)

Options

A. `P=y+x`
B. `P=y-x`
C. `P+Q=1-x+y+y^(')+(y^('))^(2)`
D. `P-Q=x+y-y^(')-(y^('))^(2)`

Answer

B, C

Explanation

Centers of circle lie on the straight line y=x.
`therefore` Equation of family of circles is
`(x-alpha)^(2)+(y-alpha)^(2)=r^(2)`
`therefore x^(2)+y^(2)-2alphax-2alphay+2alpha^(2)-r^(2)=0`
Differentiating w.r.t., x, we get
`2x+2yy^(')-2alphay^(')=0`.............(1)
Differentiating w.r.t. x, we get
`2x+2yy^(')-2alpha-2alphay^(')=0`
`rArr alpha=(x+yy^('))/(1+y^('))`.............(2)
Again differentiating w.r.t.x, we get
`2+2(y^('))^(2)+2yy^('')-2alphay^('')=0`
`rArr 1+(y^('))^(2)+yy^('')(x+yy^('))/(1+y^('))y^('')=0`
`rArr 1+y^(')+(y^('))^(3)+yy^('')+yy^(')y^('')-xy^('')-yy^(')y^('')=0`
`rArr (y-x)y^('')+(1+y^(')+y^('^(2))y^(2))+1=0`
`rArr P=y-x, Q=1 + y^(')+(y^('))^(2)`