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Question

If , for some prove that is a constant independent of a and b .

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Solution

It is given that ( xa ) 2 + ( yb ) 2 = c 2 .

We have to prove that [ 1+ ( dy dx ) 2 ] 3 2 d 2 y d x 2 is a constant independent of aand b.

Let the equation ( xa ) 2 + ( yb ) 2 = c 2 and differentiate it with respect to x.

d dx { ( xa ) 2 + ( yb ) 2 }= d dx ( c 2 ) 2 ( xa ) ( 21 ) d dx ( xa )+2 ( yb ) ( 21 ) d dx ( yb )=0 2( xa )+2( yb )( dy dx )=0 dy dx = ( xa ) ( yb )

Differentiate the above equation with respect to x.

d dx ( dy dx )= d dx { ( xa ) ( yb ) } d 2 y d x 2 =[ ( yb ) d dx ( xa ) d dx ( yb )×( xa ) ( yb ) 2 ] =[ ( yb )×1( dy dx )×( xa ) ( yb ) 2 ] =[ ( yb )×1 ( x+a ) ( yb ) ×( xa ) ( yb ) 2 ]

Further simplify.

d 2 y d x 2 =[ ( yb ) 2 + ( xa ) 2 ( yb ) 2 ×( yb ) ] d 2 y d x 2 = c 2 ( yb ) 3

Substitute the value of dy dx , d 2 y d x 2 in [ 1+ ( dy dx ) 2 ] 3 2 d 2 y d x 2 .

[ 1+ ( dy dx ) 2 ] 3 2 d 2 y d x 2 = [ 1+ { ( xa ) ( yb ) } 2 ] 3 2 { c 2 ( yb ) 3 } = [ ( yb ) 2 + ( xa ) 2 ( yb ) 2 { c 2 ( yb ) 3 } ] 3 2 = [ c 2 ( yb ) 2 ] 3 2 { c 2 ( yb ) 3 }

Further simplify.

[ 1+ ( dy dx ) 2 ] 3 2 d 2 y d x 2 = [ c 2 ( yb ) 2 ] 3 2 × ( yb ) 3 c 2 = ( c yb ) 2× 3 2 × ( yb ) 3 c 2 =( c 3 c 2 )× ( yb ) 3 ( yb ) 3 =c

Hence the required condition proved.


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