Atelescope contains both a parabolic mirror and a hyperbolic mirror. they share focus upper f 1, which is 46 feet above the vertex of the parabola. the hyperbola's second focus upper f 2 is 6 ft above the parabola's vertex. the vertex of the hyperbolic mirror is 2 ft below upper f 1. find the equation of the hyperbola if the center is at the origin of a coordinate system and the foci are on the y-axis.

if the center is at the origin of a coordinate system and the foci are on the y-axis, then the foci are symmetric about the origin.

The hyperbola focus F1 is 46 feet above the vertex of the parabola and the hyperbola focus F2 is 6 ft above the parabola's vertex. Then the distance F1F2 is 46-6=40 ft.

In terms of hyperbola, F1F2=2c, c=20.

The vertex of the hyperba is 2 ft below focus F1, then in terms of hyperbola c-a=2 and a=c-2=18 ft.

Use formula to find b:

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The branches of hyperbola go in y-direction, so the equation of hyperbola is

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Substitute a and b:

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