Question #60216

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Answer 1 · 2016-07-23T06:44:52

$256/625$

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)