Question 743731
Find the equation of the hyperbola that the transverse axis is parallel to the x-axis, center at (2,-2), passing through (2+3√2,0) and (2+3√10,4).
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This is a hyperbola with horizontal transverse axis.
Its standard form of equation: {{{(x-h)^2/a^2-(y-k)^2/b^2=1}}}, (h,k)=(x,y) coordinates of center
For given hyperbola:
Given center: (2,-2)
Equation: {{{(x-2)^2/a^2-(y+2)^2/b^2=1}}}
using given coordinates to solve for a and b:
(2+3√2-2)^2/a^2-(0+2)^2/b^2=1
(2+3√10-2)^2/a^2-(4+2)^2/b^2=1
..
18/a^2-4/b^2=1
90/a^2-36/b^2=1
let x=1/a^2
let y=1/b^2
..
18x-4y=1
90x-36y=1
..
-162x+36y=-9
90x-36y=1
add
-72x=-8
x=8/72=1/9=1/a^2
a^2=9
..
4y=18x-1=2-1=1
y=1/4=1/b^2
b^2=4
Equation of given hyperbola:
 {{{(x-2)^2/9-(y+2)^2/4=1}}}
..