13.7. EXERCISES 283

18. Give a simple description using the distance formula of the set of points which are atan equal distance between the two points (x1,y1,z1) and (x2,y2,z2) .

19. Suppose you are given two points (−a,0) and (a,0) in R2 and a number r > 2a. Theset of points described by{

(x,y) ∈ R2 : |(x,y)− (−a,0)| + |(x,y)− (a,0)|= r}

is known as an ellipse. The two given points are known as the focus points of the

ellipse. Find α and β such that this is in the form( x

α

)2+(

yβ

)2= 1. This is a nice

exercise in messy algebra.

20. Suppose you are given two points (−a,0) and (a,0) in R2 and a number r < 2a. Theset of points described by{

(x,y) ∈ R2 : |(x,y)− (−a,0)| −|(x,y)− (a,0)|= r}

is known as hyperbola. The two given points are known as the focus points of the

hyperbola. Simplify this to the form( x

α

)2 −(

yβ

)2= 1. This is a nice exercise in

messy algebra.

21. Let (x1,y1) and (x2,y2) be two points in R2. Give a simple description using thedistance formula of the perpendicular bisector of the line segment joining these twopoints. Thus you want all points (x,y) such that |(x,y)− (x1,y1)|= |(x,y)− (x2,y2)| .

22. Show that |αx| =|α||x| whenever x ∈ Rp for any positive integer p.