Question

PLEASE HELP I NEED TO SHOW MY WORK!!

Based on the lengths
of the segments in the
hyperbola below, how
long is the dashed
line?

Answer 1

Like the ellipse, the hyperbola can also be defined as a set of points in the coordinate plane. A hyperbola is the set of all points

(

x

,

y

)

in a plane such that the difference of the distances between

(

x

,

y

)

and the foci is a positive constant.

Notice that the definition of a hyperbola is very similar to that of an ellipse. The distinction is that the hyperbola is defined in terms of the difference of two distances, whereas the ellipse is defined in terms of the sum of two distances.

Answer 2

10

Explanation:

it is one of the properties and derived "abilities" of a hyperbola that the distance of a point on one branch to the focal point of the other branch is the sum of the minimum distance of both branches (in our case 4) and the distance to its "own" focal point (in our case 6).

so, the length of the dotted line is

4 + 6 = 10

to extend and clarify the information from the other answer :

there are 3 different curves when slicing a double-cone (2 cones standing straight on top of each other with their tips touching - in other words, each cone is the mirrored upside-down version of the other).

1. ellipse

a plane is slicing through one of the cones, so that it never touches the other cone.

the ellipse is the outline of that cut around the cone.

a circle is an extreme case, where the plane cuts perpendicular to the central axis of the cone.

2. parabola

an infinite ellipse (another extreme case of ellipse). the slicing plane is parallel to the outline of the cone(s).

but still, only one of the 2 cones is touched.

3. hyperbola

the slicing plane is so inclined that it cuts through both cones. the 2 branches of a hyperbola are the outlines of the cuts through both cones.

usually shown as example is the extreme case, where the slicing plane is parallel to the central axis of the double-cone.

all 3 forms are therefore related.