The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the l…

Question

The eccentricity of the hyperbola whose length of the latus rectum is equal to $8$ and the length of its conjugate axis is equal to half of the distance between its foci, is:

Accepted Answer

$\dfrac{2b^2}a=8\Rightarrow b^2=4a$. $2b=\dfrac12(2ae)\Rightarrow4b^2=a^2e^2$.

$$4(4a)=a^2e^2,\qquad b^2=a^2(e^2-1)\Rightarrow4a=a^2(e^2-1)\Rightarrow4=a(e^2-1).$$

Combining with $4b^2=a^2e^2$ and $b^2=4a$: $16a=a^2e^2\Rightarrow16=ae^2$. Dividing: $\dfrac{16}4=\dfrac{ae^2}{a(e^2-1)}=\dfrac{e^2}{e^2-1}\Rightarrow4e^2-4=e^2\Rightarrow3e^2=4\Rightarrow e=\dfrac2{\sqrt3}.$

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