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  4. If the latus rectum of a hyperbola subtends right angle at its centre, then its eccentricity is

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Q. If the latus rectum of a hyperbola subtends right angle at its centre, then its eccentricity is

Conic Sections

Solution:

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$\Theta $ Slope of $OL =1$
$\Rightarrow ae =\frac{ b ^2}{ a } $
$\Rightarrow a ^2 e = a ^2\left( e ^2-1\right) $
$\Rightarrow e ^2- e -1=0$
$\Rightarrow e =\frac{1 \pm \sqrt{5}}{2} $
$\therefore \text { eccentricity } =\frac{\sqrt{5}+1}{2}$