If the tangent at $\left(x_1,y_1\right)$ to the curve $x^3+y^3=a^3$ meets the curve again at $\left(x_2,y_2\right)$ then

$\frac{dy}{dx}=-\frac{x_1^2}{y_1^2}$

Tangent equation is $x_1^{2}x+y_1^{2}y=x_1^3+y_1^3\Rightarrow x_1^{2}x+y_1^{2}y=a_{ }^3$

Since, it passes through $\left(x_2,y_2\right)$

          $\therefore x_1^2{x}_2+y_1^2{y}_2=a_{ }^3\left(\text{1}\right)$
and     $x_1^3+y_1^3=a_{ }^3\left(\text{2}\right)$    
            $x_2^3+y_2^3=a_{ }^3\left(\text{3}\right)$
By solving (1), (2), (3) we get result