Question

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

Answer 1

Slope of tangent at $(x_1,y_1)$ is $-\frac{x_1^2}{y_1^2}$
The tangent cuts the curve again at $(x_2,y_2)$. Therefore,
slope of the tangent $=\frac{y_2-y_1}{x_2-x_1}$
$\Rightarrow -\frac{x_1^2}{y_1^2}=\frac{y_2-y_1}{x_2-x_1}$ $\left(1\right)$
Also, $x_1^3+y_1^3=a^3$ and $x_2^3+y_2^3=a^3$
$\therefore x_1^3+y_1^3=x_2^3+y_2^3$
$\Rightarrow \frac{y_2^3-y_1^3}{x_1^3-x_2^3}=1$
$\Rightarrow \frac{y_2-y_1}{x_1-x_2}=\frac{x_1^2+x_2^2+x_1{x}_2}{y_1^2+y_2^2+y_1{y}_2}$
$\Rightarrow \frac{x_1^2}{y_1^2}=\frac{x_1^2+x_2^2+x_1{x}_2}{y_1^2+y_2^2+y_1{y}_2}$
$\Rightarrow x_1^2{y}_1^2+x_1^2{y}_2^2+x_1^2{y}_1{y}_2=y_1^2{x}_1^2+x_2^2{y}_1^2+y_1^2{x}_1{x}_2$
$\Rightarrow x_2^2{y}_1^2-y_2^2{x}_1^2=x_1{y}_1[x_1{y}_2-x_2{y}_1]$
$\Rightarrow x_2{y}_1+y_2{x}_1=-x_1{y}_1$
$\Rightarrow \frac{x_2}{x_1}+\frac{y_2}{y_1}=-1$