Tangent at a point ${\text{P}}_1$ [other than (0, 0)] on the curve $\mathrm{y}={\mathrm{x}}^3$ meets the curve again at P₂. The tangent at P₂ meets the curve again at P₃ and so on. Then the ratio $\left(\frac{\text{ area of }{\mathrm{\Delta P}}_2{\mathrm{P}}_3{\mathrm{P}}_4}{\text{ area of }{\mathrm{\Delta P}}_1{\mathrm{P}}_2{\mathrm{P}}_3}\right)=4\mathrm{k}$. Then k =

$$\begin{gathered} \text{ Let a point }{P}_1\text{ on }y=x^3\text{ be }\left(h,h^3\right) \\ \text{ Tangent at }{P}_1\text{ is }y-h^3=3h^2\left(x-h\right)\text{, it meets }y=x^3\text{ at }{P}_2\text{. } \\ \begin{array}{l}\therefore x^3-h^3=3h^2(x-h)\\ \Rightarrow (x-h)\left(x^2+xh+h^2\right)=3h^2(x-h)\left[\because a^3-b^3=(a-b)\left(a^2+ab+b^2\right)\right]\\ \Rightarrow x^2+xh+h^2-3h^2\\ \Rightarrow x^2+xh-2h^2=0\\ \Rightarrow (x-h)(x+2h)=0\\ \therefore P_2\text{ is }\left(-2h,-8h^3\right)\text{,}{P}_3\text{ is }\left(4h,64h^3\right)\\ \end{array} \end{gathered}$$

$\begin{array}{rcl}\mathrm{ar}\left(\mathrm{\Delta }\left({\mathrm{P}}_1{\mathrm{P}}_2{\mathrm{P}}_3\right)\right)& =& \frac{1}{2}\left|\begin{array}{ccc}\mathrm{h}& {\mathrm{h}}^3& 1\\ -2\mathrm{h}& -8{\mathrm{h}}^3& 1\\ 4\mathrm{h}& 64{\mathrm{h}}^3& 1\end{array}\right|\\ \mathrm{ar}\left(\mathrm{△}\left({\mathrm{P}}_2{\mathrm{P}}_3{\mathrm{P}}_4\right)\right)& =& \frac{1}{2}\left|\begin{array}{ccc}-2\mathrm{h}& -8{\mathrm{h}}^3& 1\\ 4\mathrm{h}& 64{\mathrm{h}}^3& 1\\ -8\mathrm{h}& -512{\mathrm{h}}^3& 1\end{array}\right|\\ & =& \frac{1}{2}\left(-2\right)\left(-8\right)\left|\begin{array}{ccc}\mathrm{h}& {\mathrm{h}}^3& 1\\ -2\mathrm{h}& -8{\mathrm{h}}^3& 1\\ 4\mathrm{h}& 64{\mathrm{h}}^3& 1\end{array}\right|\\ & =& 16 \mathrm{ar}\left(\mathrm{△}\left({\mathrm{P}}_1{\mathrm{P}}_2{\mathrm{P}}_3\right)\right)\\ & \Rightarrow & \frac{\mathrm{ar}\left(\mathrm{△}\left({\mathrm{P}}_2{\mathrm{P}}_3{\mathrm{P}}_4\right)\right)}{\mathrm{ar}\left(\mathrm{△}\left({\mathrm{P}}_1{\mathrm{P}}_2{\mathrm{P}}_3\right)\right)}=16\end{array}$