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:

.

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

.

Substitute a and b:

.

x=-2

infinite number of lines, so there is an infinite number of equations.

step-by-step explanation:

there are an infinite number of line that passes through the point (-2,5)

x=-2   is one line that passes through the point

y = 5 is another line that passes through the point

(these two lines are perpendicular to each other).

since the number of   numbers are infinite and the slope is the ratio of two numbers,(change in y over change in x) we can have an infinite number of slopes.     our only condition is that it passes through the point (-2,5)