NAME_____________________________________________________DATE________________
GEOMETRY
LOCUS
Answer all questions. Show all diagrams leading to the answer on loose leaf.
1. Which equation represents the locus of points equidistant from the points (1,1) and (7,1)?
(1) x = 4
(2) y = 4
(3) x = -4
(4) y = -4
2. If the graphs of the equations x2 + y2 = 25 and y = x are drawn on the same set of axes, what is the total
number of points common to both graphs?
(1) 1
(2) 2
(3) 3
(4) 0
3. What is the locus of points at a given distance from a given line?
(1) 1 point
(2) 2 points
(3) 1 circle
(4) 2 parallel lines
4. The equation of the locus of points 5 units from the origin is:
(1) x2 + y2 = 5
(2) x2 + y2 = 25
(3) x = 5
(4) y = 5
5. An intersection of the graphs of the equations y = -x and y = x2  2 is
(1) (-1,-1)
(2) (-2,2)
(3) (2,2)
(4) (1,1)
6. Which is an equation of the locus of points equidistant from the points (-2,0) and (4,0)?
(1) x = 1
(2) x = -1
(3) y = 1
(4) y = -1
7. When drawn on the same set of axes, which pair of equations will result in two points of intersection?
(1) y = x and y = -x (2) y = 3 and x = -3 (3) y = x2 and y =-x2
(4)y = x and x2 + y2 = 1
8. Which equation represents the locus of points equidistant from the line whose equations are
and y = 3x  6?
(1) y = 3x  1
(2) y = 3x + 1
(3) y = 3x  4
(4) y = 3x + 4
y = 3x + 8
9. Which equation represents the locus of points equidistant from the points (4,2) and (8,2)?
(1) x = 6
(2) y = 6
(3) x = 12
(4) y = 12
10. If point P is on line l, what is the total number of points 3 centimeters from point P and 4 cm from line l?
(1) 1
(2) 2
(3) 0
(4) 4
11. If the graphs of the equations x2 + y2 = 16 and y = 4 are drawn on the same set of axes, what is the total
number of points common to both graphs?
(1) 1
(2) 2
(3) 3
(4) 0
12. Lines l and m are parallel lines 8 cm apart and point P is on line l. What is the total number of points that
are equidistant from lines l and m and 5 cm from P?
(1) 1
(2) 2
(3) 0
(4) 4
NAME_______________________________________________DATE_____________
GEOMETRY
LOCUS (REGENTS QUESTIONS)
Draw a diagram for each of the following on graph paper or loose leaf.
(Homework will be unacceptable if not accompanied with diagrams.)
PART I:
1. The locus of the midpoints of the radii of a circle is
(1) a point
(2) two lines
(3) a line
(4) a circle
2. How many points are equidistant from two intersecting lines, l and m, and 3 units from the point
of intersection of the lines?
(1) 1
(2) 2
(3) 3
(4) 4
3. The distance between points P and Q is six units. How many points are equidistant from P and Q
and also three units from P?
(1) 1
(2) 2
(3) 3
(4) 6
4. The distance between points P and Q is 8 units. How many points are equidistant from P and Q
and also 3 units from P?
(1) 1
(2) 2
(3) 0
(4) 4
5. The distance between parallel lines l and m is 12 units. Point A is on line l. How many points are
equidistant from lines l and m and 8 units from point A?
(1) 1
(2) 2
(3) 3
(4) 4
6. In the coordinate plane, what is the total number of points 5 units from the origin and equidistant
from both the x- and y-axes?
(1) 1
(2) 2
(3) 0
(4) 4
7. How many points are equidistant from two intersecting lines and 3 units from their point of
intersection?
(1) 1
(2) 2
(3) 3
(4) 4
8. An equation of the locus of points equidistant from the points (0,6) and (0,-2) is
(1) x = 2
(2) y = 2
(3) x = -2
(4) y = -2
9. If the graphs of x2 + y2 = 4 and y = -4 are drawn on the same set of axes, what is the total number
of points common to both graphs?
(1) 1
(2) 2
(3) 3
(4) 0
10. Which is an equation of the locus of points that are 3 units from the point (3,2)?
(1) (x  3)2 + (y  2)2 = 9
(3) (x  3)2 + (y  2)2 = 3
(2) (y  2) = (x  3) + 3
(4) y = 3x  7
11. What is the total number of points that the graphs of x2 + y2 = 16 and y = x have in common?
(1) 1
(2) 2
(3) 0
(4) 4
12. Points M and N lie on line l. Line k is parallel to line l. The total number of points equidistant
from points M and N and also equidistant from lines l and k is
(1) 1
(2) 2
(3) 0
(4) 4
PART II:
1. Maria's backyard has two trees that are 40 feet apart, as shown in the accompanying diagram. She
wants to place lampposts so that the posts are 30 feet from both of the trees. How many locations
for the lampposts are possible?
2. A treasure map shows a treasure hidden in a park near a tree and a statue. The map indicates that
the tree and the statue are 10 feet apart. The treasure is buried 7 feet from the base of the tree and
also 5 feet from the base of the statue. How many places are possible locations for the treasure to
be buried? Draw a diagram of the treasure map, and indicate with an X each possible location of
the treasure.
3. An architect is deciding where to install a fountain in a long rectangular courtyard. The fountain
should be equidistant from the north and south sides of the courtyard. It should also be 30 feet
away from the entrance on the north side. The courtyard is 60 feet across. Create a map showing
where the fountain should be installed.
Map of the Courtyard
North Entrance
60 ft.
South Entrance
NAME_____________________________________________DATE_______________
GEOMETRY
LOCUS
In 1-3: a) On graph paper, draw the locus of points equidistant from the two points.
13. Write the equation of the locus drawn.
1. (0,2) and (0,10)
2. (3,0) and (9,0)
3. (2,6) and (2,12)
In 4-6, write an equation of the locus based upon the given conditions.
4. 3 units from the line x = 5
5. 2 units from the line y = 7
6. 5 units from the line y = 1
In 7-10, the equations of two fixed lines are given. Write an equation of the locus of points
equidistant from the two fixed lines.
7. x = 4 and x =10
8. y = 3 and y = -8
9. y = x and y = x + 2
10. y = -1/2 x + 7 and y = -1/2 x + 1
11. Which is an equation of the locus of points 5 units from the y-axis?
(1) y = 5
(2) x = 5 or x = -5
(3) y = 5 or y = -5
(4) x2 + y2 = 25
a.
Which is an equation of the locus of points equidistant from two lines whose equations are y = 3x
+ 5 and y = 3x  1?
(1) y = 3x
(2) y = 3x + 2
(3) y = 3x + 4
(4) y = 3
b.
Which is an equation of the locus of points equidistant from the x-axis and the y-axis?
(1) x = y
(2) x = 0 or y = 0
(3) x = y or x = -y (4) x2 + y2 = r2
c.
a) On graph paper, draw the intersecting lines whose equations are y = 2x and
y = -2x + 4. b) Draw the locus of points equidistant from these two intersecting
lines. c) Write an equation of the locus described in part b.
d.
a. On graph paper, draw the intersecting lines whose equations are y = 1/2 x +1 and y = -1/2 x
+ 5. b) Draw the locus of points equidistant from these two intersecting lines. c) Write an
equation of the locus described in part b.
e.
Write an equation of the locus of points equidistant from (0,-3) and (0,7).
f.
a. Draw the locus of points equidistant from the points (4,1) and (4,5) and write an
equation for this locus.
b. Draw the locus of points equidistant from the points (3,2) and (9,2) and write an equation for
this locus.
c. Find the number of points that satisfy both conditions stated in a and b. Give the coordinates
of each point found.
g.
a. Represent graphically the locus of points: (1) 3 units from the line x = 1 (2) 4
units from the line y = -2
b. Write the equations for the loci represented in a.
c. Find the coordinates of the points of intersection of these loci.
h.
a. Represent graphically the locus of points: (1) 8 units from the y-axis (2) 10 units
from the origin
b. Write equations for the loci represented in a.
c. Find the coordinates of the points of intersection of these loci.
i.
a. Draw the locus of points equidistant from the circles whose equation are
x2 + y2 = 4 and x2 + y2 = 36. Write an equation of the locus.
b. Draw the locus of points 4 units from the x –axis. Write an equation of the locus.
c. Find the number of points that satisfy both conditions stated in a and b. Write the coordinates
of each of the points found.