66 CHAPTER 5. FUNDAMENTALS
15. A parabola is the set of all points (x,y) in the plane such that the distance from thepoint (x,y) to a given point (x0,y0) equals the distance from (x,y) to a given line.The point (x0,y0) is called the focus and the line is called the directrix. Find theequation of the parabola which results from the line y = l and (x0,y0) a given focuswith y0 < l. Repeat for y0 > l.
16. A sphere centered at the point (x0,y0,z0) ∈ R3 having radius r consists of all points(x,y,z) whose distance to (x0,y0,z0) equals r. Write an equation for this sphere inR3.
17. Suppose the distance between (x,y) and (x′,y′) were defined to equal the larger ofthe two numbers |x− x′| and |y− y′| . Draw a picture of the sphere centered at thepoint (0,0) if this notion of distance is used.
18. Repeat the same problem except this time let the distance between the two points be|x− x′|+ |y− y′| .
19. If (x1,y1,z1) and (x2,y2,z2) are two points such that |(xi,yi,zi)|= 1 for i = 1,2, showthat in terms of the usual distance,
∣∣( x1+x22 , y1+y2
2 , z1+z22
)∣∣< 1. What would happen ifyou used the way of measuring distance given in Problem 17 (|(x,y,z)|= maximumof |z| , |x| , |y| .)?
20. 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) .
21. 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.
22. 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.
23. 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)| .
24. Show that |αx| =|α||x| whenever x ∈ Rn for any positive integer n.