General equation of midpoint of chord of hyperbola

Question

$ \frac{x^2}{a^2}+ \frac{y^2}{b^2} =1$

is

$ \frac{xx'} {a^2} + \frac{yy'} {b^2} = \frac{x'^2}{a^2}+ \frac{y'^2}{b^2}$

where (x',y') are coordinates of midpoint

This is apparently true for all conics. Where is this coming from?

Answer 1

From this book. For more on Joachimsthal's notation see this page.

Find the equation of the chord of $s=0$ whose midpoint is $P_1$.

Let $AA'$ be the chord and if possible let it meet the polar of $P_1$ in $P_2$. Then $AP_1=P_1A'$, $$AP_2=-P_2A'=A'P_2,$$ which is absurd. Therefore the chord cannot meet the polar of $P_1$. Hence it is parallel to the polar and has equation $s_1=k$. But it passes through $P_1$.

$\therefore$ $s_{11}=k$. Hence the chord is $$s_1=s_{11}.$$