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Question

The equation of the hyperbola whose foci are at (4,6), (4,−4) respectively and having eccentricity 2 is

A
(y1)225/8(x4)275/4=1
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B
(y1)225/4(x4)275/4=1
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C
(y1)275/4(x4)225/4=1
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D
(y1)275/4(x4)225/8=1
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Solution

The correct option is B (y1)225/4(x4)275/4=1
The centre of the hyperbola is the midpoint of the line joining the two foci. So the coordinates of the centre are (4+42,642) i.e. (4,1)
Since here, the x coordinate for foci is same, therefore transverse axis lies parallel to y axis.
The equation of hyperbola with centre at (4,1) is (y1)2a2(x4)2b2=1
Distance between the foci =2ae
(44)2+(6+4)2=2a×2
10=4aa=52
b2=a2(e21)=254(41)=754
Thus the equation of the hyperbola is (y1)225/4(x4)275/4=1

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