1.

If L_(T')L_(N')L_(ST) and L_(SN) denote the lengths of tangent, normal sub-tangent and sub-normal, respectively, of a curve y = f(x) at a point P(2009, 2010) on it, then

Answer»

`(L_(ST))/(2010)=(2010)/(L_(SN))`
`|(L_(T))/(L_(N))sqrt((L_(SN))/(L_(ST)))|="constant"`
`1-L_(ST)L_(SN)=(2000)/(2010)`
`((L_(T)+L_(N))/(L_(T)-L_(N)))^(2)=(L_(ST))/(L_(SN))`

SOLUTION :`L_(ST)=|(y)/(m)|,L_(SN)=|ym|`
`L_(T)=|(4sqrt(1+m^(2)))/(m)|.L_(N)=|ysqrt(1+m^(2))|`
where `m=(DY)/(dx)` at point `P=(x,y)` on the CURVE `y = f(x)`
Now `(L_(ST))/(L_(SN))=(1)/(m^(2))=((L_(T))/(L_(N)))^(2) and L_(ST)L_(SN)=y^(2)`


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