Which of the following is INCORRECT for the hyperbola x2−2y2−2x+8y−1=0
A
Its eccentricity is √2
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B
Length of the transverse axis is 2√3
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C
Length of the conjugate axis is 2√6
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D
Latus rectum is 4√3
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Solution
The correct option is A Its eccentricity is √2 The equation of the hyperbola is x2−2y2−2x+8y−1=0 or (x−1)2−2(y−2)2+6=0 or (x−1)2−6+(y−2)23=1 or (y−2)23−(x−1)26=1....(1) or Y23−X26=1 where X=x−1 and Y=y−2...(2) ∴ The centre =(0,0) in the X−Y co-ordinates. ∴ The centre =(1,2) in the x−y co-ordinates, using (2). If the transverse axis be of length 2a, then a=√3, since in the equation (1) the transverse axis is parallel to the y-axis. If the conjugate axis is of length 2b, then b=√6. But b2=a2(e2−1) ∴6=3(e2−1),∴e2=3 or e=√3 The length of the transverse axis =2√3. The length of the conjugate axis =2√6. Latus rectum =2b2a=2×6√3=4√3.