The correct option is B [1+(dydx)2]3=a2(d2ydx2)2
Given differential eqn is
(x−h)2+(y−k)2=a2 .....(1)
Differentiating (1) w.r.t. x ,
2(x−h)+2(y−k)dydx=0
(x−h)+(y−k)dydx=0 ....(2)
⇒dydx=−x−hy−k ....(3)
Squaring both sides of eqn (3), we get
(dydx)2=(x−h)2(y−k)2
Adding 1 to both sides, we get
⇒1+(dydx)2=a2(y−k)2
⇒(y−k)2=a21+(dydx)2 ....(4)
Differentiating (2) w.r.t. x, we get
1+(y−k)d2ydx2+(dydx)2=0
Substituting the value of y+k from (4), we get
1+a√1+(dydx)2d2ydx2+(dydx)2=0
⇒1+(dydx)2=−a√1+(dydx)2d2ydx2
⇒(1+(dydx)2)3/2=−ad2ydx2
Squaring both sides, we get
[1+(dydx)2]3=a2(d2ydx2)2