Question

Question #60216

Answer 1

`256/625`

Explanation:

Let the`P (h,k)` is any point on the curve `4x^5=5y^4`

Hence this coordinate will satisfy the equation of the curve,
So we have

`4h^5=5k^4=>k^4/h^5=4/5.........(1)`

Now differentiating equation of the given curve w.r.to x we ge

`4*5x^4=5*4y^3(dy)/(dx)=>(dy)/(dx)=x^4/y^3`

If m is the slope of the tangent to the curve at `P(h,k)` then

`m=((dy)/(dx))_(h,k)=h^4/k^3`

Now

`"Length of subtangent at (h,k) "(T) =k/m=k^4/h^4`

`"Length of subnormal at (h,k) "(N) =k*m=h^4/k^2`

`"So the required ratio"=T^3/N^2=(k^4/h^4)^3/(h^4/k^2)^2=k^12/h^12*k^4/h^8`

`=k^16/h^20=(k^4/h^5)^4=(4/5)^4=256/625`

Inserting `k^4/h^5=4/5` from (1)