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Find the vertices, eccentricity, foci, equation of directrices length of latus rectum for the hyperbola `3y^(2)-x^(2)=27`.

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`3y^(2)-x^(2)=27`.
`rArr" "(y^(2))/(9)-(x^(2))/(27)-1`
which is the equation of conjugate hyperbola
Here `b^(2)=9anda^(2)=27`
`rArr""b=3anda=3sqrt(3)`
Co-ordinates of vertices `(0,pmb)=(0,pm3)`.
Eccentricity e `=sqrt(1+(a^(2))/(b^(2)))`
`=sqrt(1+(27)/(9))=2`.
Co-ordinates of foci `=(0,pmbe)`
`=(0,pm3xx2)`
`(0,pm6)`.
Equation of directrices `y-pm(b)/(e)" "rArr" "y=pm(3)/(2)`.
Length of latus rectum `=(2a^(2))/(b)=(2xx27)/(3)=18`.
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