Prove that DE is parallel to BC.
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see explanation

Step-by-step explanation:

Parallel lines have equal slopes.

To find D and E use the midpoint formula

Given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is

( , )

Here (x₁, y₁ ) = A(4, 6) and (x₂, y₂ ) = B(2, - 2) , then

D = (, ) = (3, 2 ) and

let (x₁, y₁ ) = B(2, - 2 ) and (x₂, y₂ ) = C(- 2, - 4 ), then

E = ( , ) = (1, 1 )

Use the slope formula to find slopes of DE and BC

m =

with (x₁, y₁ ) = D(3, 2) and (x₂, y₂ ) = E(1, 1), then

= = =

Repeat with (x₁, y₁ ) = B(2, - 2) and (x₂, y₂ ) = C(- 2, - 4), then

= = =

Since the slopes are equal then DE and BC are parallel lines

the transverse axis of a hyperbola is the line that contains the two vertices and the two focuses.

in this example (hyperbola of equation x^2/2 - y^2/4 = 1), the transverse axis is the  x-axis  . 

step-by-step explanation:

were is the equation