The tangent at any point of the curve $\mathrm{x}={\mathrm{at}}^3$, $\mathrm{y}={\mathrm{at}}^4$ divides the abscissa of the point of contact in the ratio

$$\begin{gathered} x=a{t}^3, y=a{t}^4 \\ \frac{dx}{dt}=3a{t}^2 \\ \frac{dy}{dt}=4a{t}^3 \\ \frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{4a{t}^3}{3a{t}^2} \\ =\frac{4t}{3} \end{gathered}$$
$$\begin{gathered} equation of \tan gent y-y_1=m(x-x_1) \\ y-a{t}^4=\frac{4t}{3}(x-a{t}^3) \\ 3y-3a{t}^4=4tx-4a{t}^4 \\ 4tx-3y=a{t}^4 \\ y=0\Rightarrow x=\frac{a{t}^3}{4} \\ x:x_1=\frac{a{t}^3}{4}:a{t}^3=1:4 \end{gathered}$$