The length of normal at any point to the curve, `y=c cosh(x/c)` is

Options

A. fixed
B. ` ( y^ 2 )/( c^ 2 ) `
C. ` (y ^ 2 ) /( c ) `
D. ` ( y ) /( c ^ 2 ) `

Answer

C

Explanation

Given `, y = c cos h (( x)/ ( c )) `
` ( d y ) /( dx ) = c * ( 1 )/ (c ) * sin h (( x )/ ( c )) = sin h ( ( x )/( c )) `
Now, length of normal
` = y sqrt ( 1 + (( dy )/( dx ))^ 2 ) `
` = c cos h ( ( x )/( c )) sqrt ( 1 + sin ^ 2 h (( x )/ (c )) ) `
` = c cos h (( x )/ (c )) sqrt ( cos h^ 2 ( ( x )/( c )) ) `
` = c[ cos h (( x )/ ( c))]^ 2 `
` = c (( y ) / ( c ))^ 2 " " `[from Eq. (i) ]
` = ( y^ 2 )/( c ) `