Question

Find the condition on a and b for which two distinct chords of the hyperbola $\frac{x^2}{2a^2}-\frac{y^2}{2b^2}=1$ passing through (a, b) are bisected by the line x + y = b.

Answer 1

Let the line x + y = b bisect the chord at P (a, b-a)

$\therefore$ equation of chord whose mid-point is P (a,b-a)

$\frac{xa}{2a^2}-\frac{y(b-a)}{2b^2}=\frac{a^2}{2a^2}-\frac{{(b-a)}^2}{2b^2}$

Since it passes through (a,b)

$\therefore \frac{a}{2a}-\frac{(b-a)}{2b}=\frac{a^2}{2a^2}-\frac{{(b-a)}^2}{2b^2}{a}^2(\frac{1}{a^2}-\frac{1}{b^2})+a(\frac{1}{b}-\frac{1}{a})=0a0,a-\frac{1}{\frac{1}{a}+\frac{1}{b}}\therefore a\ne \pm b$