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Question

The differential equation whose solution is (x−h)2+(y−k)2=a2 is (a is a constant ):

A
[1+(dydx)2]3=a2d2ydx2
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B
[1+(dydx)2]3=a2(d2ydx2)2
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C
[1+(dydx)]3=a2(d2ydx2)2
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D
None of these
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Solution

The correct option is B [1+(dydx)2]3=a2(d2ydx2)2
Given differential eqn is
(xh)2+(yk)2=a2 .....(1)
Differentiating (1) w.r.t. x ,
2(xh)+2(yk)dydx=0
(xh)+(yk)dydx=0 ....(2)
dydx=xhyk ....(3)
Squaring both sides of eqn (3), we get
(dydx)2=(xh)2(yk)2
Adding 1 to both sides, we get
1+(dydx)2=a2(yk)2
(yk)2=a21+(dydx)2 ....(4)
Differentiating (2) w.r.t. x, we get
1+(yk)d2ydx2+(dydx)2=0
Substituting the value of y+k from (4), we get
1+a1+(dydx)2d2ydx2+(dydx)2=0
1+(dydx)2=a1+(dydx)2d2ydx2
(1+(dydx)2)3/2=ad2ydx2
Squaring both sides, we get
[1+(dydx)2]3=a2(d2ydx2)2

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