Mathematics - Application of Derivative Question with Solution | TestHub

Let $P_i\left(x_i,y_i\right),i=1,2,3,\dots ,n$ $y=x^3(1)\Rightarrow \frac{dy}{dx}=3x^2$ Equation of tangent at $P_1\left(x_1,y_1\right):y-x_1^3=3x_1^2\left(x-x_1\right)$ Solving (1) and (2), $x=-2x_1$ $\therefore x_2=-2x_1\text{ and }{y}_2=-8x_1^3,P_2\equiv \left(-2x_1,-8x_1^3\right)$

and like wise $P_3\equiv \left(4x_1,64x_1^3\right)$ Abscissa of $P_1,P_2,\dots ,P_n$ are given by $x_1,-2x_1,4x_1,-8x_1,\dots$ which is G. P. with common ratio -2. Area of $\left(\Delta P_1{P}_2{P}_3\right)=\frac{1}{2}\left|\begin{array}{ccccccc}{x}_1& x_1^3& 1-2x_1& -8x_1^3& 14x_1& 64x_1^3& 1\end{array}\right|=\frac{x_1^4}{2}\left|\begin{array}{ccccccc}1& 1& 1-2& -8& 14& 64& 1\end{array}\right|$ Area of $\left(\Delta P_2{P}_3{P}_4\right)=\frac{1}{2}\left|\begin{array}{ccccccc}-2x_1& -8x_1^3& 14x_1& 64x_1^3& 1-8x_1& -512x_1^3& 1\end{array}\right|=8x_1^4\left|\begin{array}{ccccccc}1& 1& 1-2& -8& 14& 64& 1\end{array}\right|$ So area of $\left(\Delta P_2{P}_3{P}_4\right)=16$ area of ( $\Delta P_1{P}_2{P}_3$)