# Question #60216

Jul 23, 2016

$\frac{256}{625}$

#### Explanation:

Let the$P \left(h , k\right)$ is any point on the curve $4 {x}^{5} = 5 {y}^{4}$

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

$4 {h}^{5} = 5 {k}^{4} \implies {k}^{4} / {h}^{5} = \frac{4}{5.} \ldots \ldots . . \left(1\right)$

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

$4 \cdot 5 {x}^{4} = 5 \cdot 4 {y}^{3} \frac{\mathrm{dy}}{\mathrm{dx}} \implies \frac{\mathrm{dy}}{\mathrm{dx}} = {x}^{4} / {y}^{3}$

If m is the slope of the tangent to the curve at $P \left(h , k\right)$ then

$m = {\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)}_{h , k} = {h}^{4} / {k}^{3}$

Now

$\text{Length of subtangent at (h,k) } \left(T\right) = \frac{k}{m} = {k}^{4} / {h}^{4}$

$\text{Length of subnormal at (h,k) } \left(N\right) = k \cdot m = {h}^{4} / {k}^{2}$

$\text{So the required ratio} = {T}^{3} / {N}^{2} = {\left({k}^{4} / {h}^{4}\right)}^{3} / {\left({h}^{4} / {k}^{2}\right)}^{2} = {k}^{12} / {h}^{12} \cdot {k}^{4} / {h}^{8}$

$= {k}^{16} / {h}^{20} = {\left({k}^{4} / {h}^{5}\right)}^{4} = {\left(\frac{4}{5}\right)}^{4} = \frac{256}{625}$

Inserting ${k}^{4} / {h}^{5} = \frac{4}{5}$ from (1)