Thread: If e and e' are eccentricities of hyperbola

1. If e and e' are eccentricities of hyperbola

Question :
If e and e' are eccentricities of hyperbola and its conjugate, prove that

$
\left(\frac{1}{e^2} \right) + \left( \frac{1}{e'^2} \right) = 1
$

2. Hello, zorro!

If $e$ and $e'$ are eccentricities of a hyperbola and its conjugate,

prove that: . $\frac{1}{e^2}+ \frac{1}{(e')^2} \:=\: 1$

A hyperbola is: . $\frac{x^2}{a^2} - \frac{y^2}{b^2} \:=\:1$
. . Its eccentricity is: . $e \:=\:\frac{c}{a}\:=\:\frac{\sqrt{a^2+b^2}}{a}$

Its conjugate is: . $\frac{y^2}{b^2} - \frac{x^2}{a^2} \:=\:1$
. . Its eccentricity is: . $e' \:=\:\frac{c}{b}\:=\:\frac{\sqrt{a^2+b^2}}{b}$

$\text{Hence: }\;\frac{1}{e^2} + \frac{1}{(e')^2} \;\;=\;\;\frac{1}{\left(\frac{\sqrt{a^2+b^2}}{a}\r ight)^2} + \frac{1}{\left(\frac{\sqrt{a^2+b^2}}{b}\right)^2}$

. . . $=\;\;\frac{1}{\frac{a^2+b^2}{a^2}} + \frac{1}{\frac{a^2+b^2}{b^2}} \;\;=\;\; \frac{a^2}{a^2+b^2} + \frac{b^2}{a^2+b^2} \;=\;\frac{a^2+b^2}{a^2+b^2} \;\;=\;\;1$

3. Thanks u very much

Thanks u very much every one for helping me

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if e an e' be the ecentricities of ahyperbola and its conjugate prove

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