Transverse axis parallel to the x-axis,center at (2, -2) and passing through (2+3 square root of 2, 0) and (2+3 square root of 10, 4). Find the equation of hyperbola.
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If the transverse axis is parallel to x-axis or horizontal, the form of the equation of hyperbola is,
`(x-h)^2/a^2 - (y-k)^2/b^2 = 1`
(The transverse axis of a hyperbola is the line which connects the two vertices through the center of the hyperbola)
(Therefore this is an east-west opening hyperbola)
Where, (h,k) is the center of the hyperbola, a is the semi major axis and b is the semi minor axis.
It is given that the center of the hyperbola is (2,-2). Therefore, h = 2 and k = -2. The equations becomes,
`(x-2)^2/a^2 - (y-(-2))^2/b^2 = 1`
`(x-2)^2/a^2 - (y+2)^2/b^2 = 1`
To find a and b, we can incorporate the other two data given.
It passes through `(2+3sqrt(2),0)` and `(2+3sqrt(10),4)` . Using above two coordinates, we can find two equations for `1/a^2` and `1/b^2` .
Using first coordinate, `(2+3sqrt(2),0)`,
`(2+3sqrt(2)-2)^2/a^2 - (0+2)^2/b^2 = 1`
`(3sqrt(2))^2/a^2 - (2)^2/b^2 = 1`
`18/a^2 - 4/b^2 = 1`
This is equation 1.
Using second coordinate, `(2+3sqrt(10),4)`
`(2+3sqrt(10)-2)^2/a^2 - (4+2)^2/b^2 = 1`
`(3sqrt(10))^2/a^2 - (6)^2/b^2 = 1`
`90/a^2 - 36/b^2 = 1`
This is equation 2,
Now we have two equations,
`18/a^2 - 4/b^2 = 1` and `90/a^2 - 36/b^2 = 1`
By solving these two simultaneous equations, you would get,
`1/a^2 = 1/9` and `1/b^2 = 1/4`
Therefore the equation of the hyperbola is,
`(x-2)^2/9 - (y+2)^2/4 = 1`
Or
`(x-2)^2/3^2 - (y+2)^2/2^2 = 1`
The center is at (2,-2), semi major axis is 3 and semi minor axis is 2.
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