Examples for Locus of Points – Key without Sketches
Describe the locus of points that satisfy each set of conditions. Describe algebraically with
an equation those that are starred. (Some may not have meaning is all three settings.)
1 Dimension
2 Dimensions
3 Dimensions (Space)
Condition
(on a Line)
(on a Plane)
Be prepared to
Draw the figure.
Draw the figure.
model it.
2
2
1. 3 units from the
Two points
x +y =9
x2 + y2 + z2 = 9
origin*
A(3) and B(-3)
circle with center at sphere with center at
the origin and with
the origin and with
radius 3
radius 3
2. Equidistant from
two points P and
Q*
From P(2) and Q(8)
midpoint at
M(5)
From P(2,0) and
Q(8,0)
From P(2,0,0) and
Q(8,0,0)
line which is the
perpendicular
bisector of PQ
plane which is
perpendicular to the
x axis at (5, 0, 0)
plane x = 5
Plane perpendicular
to the x axis at
(4.5, 0, 0)
3. 5 units away from
x = 2*
two points A(-3)
and B(7)
line x = 5
Two lines with
equations x = -3 and
x=7
4. Equidistant from
x = 3 and x = 6*
Point M(4.5)
Line x = 4.5
Two planes with
equations x = -3 and
x=7
x = 4.5
5. Equidistant from
x = 0 and y = 0*
No meaning
Two lines which
bisect the vertical
angles formed by the
x and y axes.
Lines y = x and y = -x
Two planes which
bisect the dihedral
vertical angles
formed by the planes
x = 0 and y = 0
Planes y = x and y = -x
6. Equidistant from
x = 2 and y = 4*
No meaning
Two lines passing
through (2, 4) with
slope 1 and with slope
-1
Two planes bisecting
the dihedral vertical
angles formed by the
planes x = 2 and y = 4
y = x + 2 and
y = -x + 6
Circle with
center at the origin
and radius 6
y = x + 2 and
y = -x + 6
Baseless cylinder
with axis of the
cylinder being the zaxis and radius of 6
7. x2 + y2 = 36
No meaning
8. 2 units from
x2 + y2 = 36
No meaning
Two circles with
center at the origin
and radii of 4 and 8
Two baseless
cylinders with axis of
the cylinders being
the z-axis and radii
of 4 and 8
9. 6 units from
x2 + y2 = 36
No meaning
Point at the origin
and circle with
center at the origin
and radius 12
The z-axis and
baseless cylinder with
axis the z-axis and
radius 12
10. 8 units from
x2 + y2 = 36
No meaning
Circle with center at
the origin and radius
14
Baseless cylinder
with axis being the zaxis and radius 14
11. equidistant from
x2 + y2 = 4 and
x2 + y2 = 36
No meaning
Circle with center at
the origin and radius
4
Baseless cylinder
with axis being the zaxis and radius 4
12. 5 units from the
x-axis
No meaning
Lines y = 5 and y = -5
Baseless cylinder
with axis being the xaxis and radius 5