The equation of the line passing through the points $\left(\mathrm{ct}_1,…
The equation of the line passing through the points $\left(\mathrm{ct}_1, \frac{\mathrm{c}}{\mathrm{t}_1}\right)$ and $\left(\mathrm{ct}_2, \frac{\mathrm{c}}{\mathrm{t}_2}\right)$ is
$x+t_1 t_2 y=c\left(t_1+t_2\right)$
$y+t_1 t_2 x=c\left(t_1+t_2\right)$
$x-t_1 t_2 y=c\left(t_1+t_2\right)$
$y-t_1 t_2 x=c\left(t_1+t_2\right)$
Solution
The equation of line passing through $\left(c t_1, \frac{c}{t_1}\right)$ and $\left(c t_2, \frac{c}{t_2}\right)$ is
$\begin{aligned} & \left(y-\frac{c}{t_1}\right)=\frac{\frac{c}{t_2}-\frac{c}{t_1}}{c t_2-c t_1}\left(x-c t_1\right) \\ & \Rightarrow \quad\left(y-\frac{c}{t_1}\right)=-\frac{1}{t_1 t_2}\left(x-c t_1\right) \\ & \Rightarrow \quad t_1 t_2 y-t_2 c=-x+c t_1 \\ & \Rightarrow \quad x+t_1 t_2 y=c\left(t_1+t_2\right)\end{aligned}$