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The director circle of a hyperbola is x^...

The director circle of a hyperbola is `x^(2) + y^(2) - 4y =0`. One end of the major axis is (2,0) then a focus is (a) `(sqrt(3),2-sqrt(3))` (b) `(-sqrt(3),2+sqrt(3))` (c) `(sqrt(6),2-sqrt(6))` (d) `(-sqrt(6),2+sqrt(6))`

A

`(sqrt(3),2-sqrt(3))`

B

`(-sqrt(3),2+sqrt(3))`

C

`(sqrt(6),2-sqrt(6))`

D

`(-sqrt(6),2+sqrt(6))`

Text Solution

Verified by Experts

The correct Answer is:
C, D
Radius of director circle `sqrt(a^(2)-b^(2)) =2`
Axis of hyperbola is line joining the center `C(0,2)` and `A(2,0)` (end of major axis)
`:. a = CA = 2 sqrt(2)`
`:. (2sqrt(2))^(2)-b^(2)=4`
`:. b^(2) =4`
`:. e = (sqrt(3))/(sqrt(2))`
Center of the hyperbola is center of the director circle (0,2) Focus lies on this line at distance ae from center
`:.` If foci are (x,y) then `(x-0)/(-(1)/(sqrt(2))) = (y-2)/((1)/(sqrt(2))) = +- 2 sqrt(3)`
`(x,y) = (sqrt(6),2-sqrt(6))` or `(-sqrt(6),2+sqrt(6))`
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