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Question

For the hyperbola 3y216x212y+32x52=0 the correct option(s) is/are

A
its length of transverse axis is 8 units
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B
its length of conjugate axis is 23 units
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C
end points of one latus rectum will be (14,2+19) and (74,2+19)
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D
Distance between the two directrices is 1619 units
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Solution

The correct option is C end points of one latus rectum will be (14,2+19) and (74,2+19)
3y216x212y+32x52=0
3(y2)216(x1)2=48
(y2)216(x1)23=1 ...(1)
This equation represents a hyperbola whose centre is at (1,2)
and eccentricity
e=1+(3)242=194
Here,
b2=16b=4,
a2=3, a=3.
length of transverse axis =2b=8 units
and length of conjugate axis =2a=23 units.
Distance between the two directrices =2be=3219 units

Coordinates of foci will be (1,2±be)=(1,2±19)
End points of latus rectum will be
y coordinate of latus rectum will be 2±19
Putting value of y in eqn (1), we get
x=14,74
end points of one latus rectum will be (14,2+19), (74,2+19), (14,219) and (74,219)

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