If  the tangent at  $(x_1,y_1)$  to the curve $x^3+y^3=a^3$  meets the curve again at  $(x_2,y_2)$ then  the absolute value of  $\frac{x_2}{x_1}+\frac{y_2}{y_1}=$

 $\frac{dy}{dx}=-\frac{x_1^2}{y_1^2}$
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  $(x_2,y_2)$
 $\therefore x_1^2{x}_2+y_1^2{y}_2=a^3\left(1\right)$
  $::x_1^2{X}_2+y_1^2{y}_2=a^3\left|\right)$
and   $x_1^3+y_1^3=a^3\left(2\right)$
  $x_2^3+y_2^3=a^3\left(3\right)$
By solving (1), (2), (3) we get result