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Question

The equations of the transverse and conjugate axes of a hyperbola are respectively x+2y3=0, 2xy+4=0 and their respective lengths are 2 & 23. The equation of the hyperbola is


A

25(2xy+4)235(x+2y3)2=1

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B

25(x+2y3)235(2xy+4)2=1

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C

52(2xy+4)253(x+2y3)2=1

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D

52(x+2y3)253(2xy+4)2=1

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Solution

The correct option is A

25(2xy+4)235(x+2y3)2=1


Given, 2a=2a=12

and 2b=23b=13

If we take the two axes as the new coordinate system, and the point of intersection of the axes as the new origin, then in the new coordinate system, the equation of the hyperbola is

X2a2Y2b2=1

=2X23Y2=1

Let P(x,y) be the coordinates of a point on the hyperbola in the original xy system, then

X=|2xy+4|5 and Y=|x+2y3|5

[X is the distance of a point on the hyperbola from 2xy+4=0 and Y is the distance of a point on the hyperbola from x+2y3=0]

So, the required equation is 25(2xy+4)235(x+2y3)2=1


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