Find the condition on a and b for which two distinct chords of the hyperbola x2-2a2-y2-2b2-1 passing through a- b are bisected by the line x - y - b-

Let the line x + y = b bisect the chord at P (a, b a) ∴ #160; #160;equation of chord whose mid point is P (a,b a) xa/2a^2 y ( b a )/2b^2=a^2/2a^2 ...

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Byju's Answer

Standard XII

Mathematics

Chord with a Given Mid Point : Hyperbola

Find the cond...

Question

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

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Solution

Let the line x + y = b bisect the chord at P (a, b-a)
$∴$ equation of chord whose mid-point is P (a,b-a)
$\frac{x a}{2 a^{2}} - \frac{y (b - a)}{2 b^{2}} = \frac{a^{2}}{2 a^{2}} - \frac{{(b - a)}^{2}}{2 b^{2}}$
Since it passes through (a,b)
$∴ \frac{a}{2 a} - \frac{(b - a)}{2 b} = \frac{a^{2}}{2 a^{2}} - \frac{{(b - a)}^{2}}{2 b^{2}} a^{2} (\frac{1}{a^{2}} - \frac{1}{b^{2}}) + a (\frac{1}{b} - \frac{1}{a}) = 0 a 0, a - \frac{1}{\frac{1}{a} + \frac{1}{b}} ∴ a \neq \pm b$

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