LOCUS
Locus:
The set of points that satisfies a given set of conditions
(the plural of locus is loci)
Five Basic Loci:
1. points at a set distance from a fixed point
2. points equidistant from two points
3. points at a set distance from a line
4. points equidistant from parallel lines
5. points equidistant from intersecting lines
1. Distance From a Point: The Circle
Given a fixed point O and distance d, find all points that are a distance d from point O
Locus: a circle with center O and radius d.
A circle can be defined as the set of all points (x,y) in the plane that are a fixed distance
from a given point (h,k) called the center. The fixed distance is the radius of the circle.
Equation of a circle: (x  h)2 + (y  k)2 = r2
1. Graph the locus of points 3 units from point A(1,2) and find an equation for the
graph.
2. Write an equation of a circle with its center at the origin and passing through the point
(12,5).
3. Graph the equation: (x  3)2 + (y + 2)2 = 4.
4. Describe the locus of points inside a circle of radius 3 centimeters and 2 centimeters
from the edge of the circle.
5. Two dogs, Archie and Butch, are leashed to stakes that are 4 yards apart. Their
leashes allow them to go out to a distance just under 3 yards from their stakes. Janus,
the cat, teases the dogs by sitting just outside of their reach. Where can Janus sit to
tease both dogs at the same time?
6. What equation represents the locus of points 5 units from point (1,3)?
7. Identify the center and radius of the circle (x + 1)2 + (y  3)2 = 16.
8. Describe the locus of points equidistant from two concentric circles whose radii are 8
inches and 14 inches.
(1) one concentric circle of radius 6 inches
(2) two concentric circles of radius 8 inches and 14 inches
(3) one concentric circle of radius 11 inches
(4) one concentric circle of radius 22 inches
9. Points A and B are 10 inches apart. The locus of points 8 inches from A and 4 inches
from B is:
(1) a point
(2) a line
(3) a pair of points
(4) the empty set
10. Points R and G are 10 inches apart. The locus of points 13 inches from R and 2
inches from G is:
(1) a point
(2) 2 points
(3) a circle
(4) the empty set
11. State the center and the radius of each circle:
a) x2 + y2 = 36
b) x2 + (y  1)2 = 9
12. Write the equation of a circle with center (-1,-1) and radius 3.
13. Write the equation of the circle centered at the origin that passes through (3,4).
14. Write an equation of the locus of points whose distance from the origin is 4.
15. Two circular riding paths have centers 1,000 feet apart. Both paths have a radius of
600 feet. In how many places do the paths intersect?
2. Equidistant From Two Points: The Perpendicular Bisector
Given two points, A and B, find all points equidistant from A and B.
Locus: The perpendicular bisector, CD of AB.
To Find a Perpendicular Bisector of AB in the Coordinate Plane:
 Find the midpoint of AB using the midpoint formula.
 Find the slope of AB.
 Calculate the negative reciprocal of this slope. This is the slope of the perpendicular
line.
 Use the midpoint and slope to write the equation for the perpendicular bisector.
1. Find the points equidistant from A(-2,5) and B(4,5).
2. The locus of points equidistant from any two points is:
(1) one point
(2) one line
(3) two points
(4) two lines
3. The locus of points equidistant from the four vertices of a rectangle is:
(1) the empty set
(2) a point
(3) a line
(4) a pair of points
4. What is an equation of the locus of points equidistant from points (2,3) and (-6,3)?
5. Write an equation for the locus of points equidistant from A(-1,-3) and B(-1,5).
6. Marlene is directing renovation of the town swimming center. There are two pools,
with the diving board sides of the pools 70 feet apart. Marlene wants to set up
outdoor showers at an equal distance from each diving board. Where should the
showers be placed?
3. Distance From a Line: Two Parallels Lines
Given AB and a distance d, find all points that are a distance d from the line.
Locus: CD and EF, each parallel to AB and at a perpendicular distance d from AB.
To Find a Line Parallel to a Given Line in the Coordinate Plane
 Find a point on the parallel line. (This is usually given.)
 Find the slope of the given line.
 Use the coordinates and slope to write an equation for the line.
1. Find the locus of points 2 units from the line x = 3.
2. Find the locus of points 2 units from line l and 3 units from a point on line l.
4. Equidistant from Two Parallel Lines: One Parallel Line
Given AB  CD, find all points equidistant from these lines.
Locus: MN, parallel to the two lines and midway between them.
To Find a Third Parallel Line Between Two Lines in the Coordinate Plane
 Pick one point on each given line and find the midpoint of a line drawn between
them. This point will be on the locus.
 Find the slope of the given lines.
 Using the midpoint and the slope, write an equation for the third line.
3. Find the points equidistant from the lines y = 2x + 1 and y = 2x  3.
4. Which statement describes the locus of points 2 units from the x-axis?
(1) x = 2
(2) y = -2
(3) x = 2 or x = -2
(4) y = 2 or y = -2
5. What is the total number of points that are both 2 units from the x-axis and 3 units
from the origin?
(1) 0
(2) 1
(3) 2
(4) 4
6. What is the total number of points that are both 3 units from the x-axis and 3 units
from the origin?
(1) 0
(2) 1
(3) 2
(4) 4
7. Find an equation for the locus of points 4 units from the line y = 6.
8. Write an equation for the locus of points equidistant from the lines y = 6 and y = 2.
9. Find an equation for the locus of points equidistant from the lines y = -x + 5 and
y = -x + 1.
10. How many points are 5 centimeters from a line and 8 centimeters from a point P on
the line?
11. How many points are 5 centimeters from a line and 5 centimeters from a point P on
the line?
12. How many points are 5 centimeters from a line and 3 centimeters from a point P on
the line?
13. Two points, A and B, are 8 inches apart. Find the number of points that are
equidistant from A and B and 3 inches from the line passing through A and B.
5. Equidistant from Intersecting Lines: The Angle Bisector
Given AB intersecting CD, find all points equidistant from these two intersecting
lines.
Locus: A pair of perpendicular lines, WX and YZ, that bisect the angles formed by
the intersecting lines.
1. How many points are equidistant from two intersecting lines and also 5 units from the
point of intersection?
(1) 0
(2) 2
(3) 4
(4) 5