Algebra/Loci

Algebra

The Distance Formula

Algebra Chapter 17: Conic Sections Section 2: Loci of Points

Conic Sections

17.2: Loci of Points

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Up to this point, all of the equations that we have solved or graphed had only one variable in mind.

Loci

A locus (plural: loci) is one of a set of points which satisfies a set of conditions. The usual result is a curve or a surface.

In real life, you likely heard of an object's location. As a matter of fact, the word "location" comes from the word "locus" itself. Loci define where in a plane or space that an object is located.

The photo on the right illustrates a set of points (or locus) of the headlights as traced under the condition that all vehicles follow the path of the road. In this analogy:

Set of Points: The headlight's locations as seen in the photograph
Condition: The road that the vehicles must follow
Locus: The lanes in the road as illustrated

Unfortunately, there is no general equation for finding loci. However, the following steps are typically used to determine the equation for a locus.

1. Contruct a diagram showing the given information
2. Locate several points that satisfy the rule or condition
3. Draw a curve or line using the located points
4. Write the equation

Equidistance

Example 17.1: Find the locus of points $P$ such that $P$ is equidistant from both axes. A point on the x-axis has the coordinates $(x,0)$, and a point on the y-axis has the coordinates $(0,y)$.

Practice Problems

Problem 17.1 (Equidistance) Find the locus of points $P$ such that $P$ is equidistant from $(5,-1)$ and $(3,-8)$.

Problem 17.2 (Ambulance Station) Two hospitals are located at the points (5, -1) and (2, 8). An ambulance station is to be built such that it is equidistant from the two hospitals. Determine the locus of points that are equidistant from the two hospitals.

Problem 17.3 (Locus of a Circle)

a. Verify that the points $(-3,4)$ and $(4,3)$ lie on the circle $x^{2}+y^{2}=25$.
b. Determine the locus of points equidistant from the points $(-3,4)$ and $(4,3)$.
c. Determine the relationship between the center of the circle and the locus of points equidistant from the points $(-3,4)$ and $(4,3)$.

Problem 17.4 (The Rod) A rod of lenth $l$ slides with its ends on the x-axis and y-axis. Find the locus of its midpoint.

Answer to Problem 4

$4x^{2}+4y^{2}=l^{2}$

Problem 17.5 (The Third Vertex) Two verticies of of a triangle are at $(-3,5)$ and $(1,2)$. Find the locus of the third vertex, such that the area of the triangle is 10 square units.

Problem 17.6 (Locus from Ordered Pairs) Sketch the set of ordered pairs. Then write an equation for a locus that all of the points in each set might satisfy.

a. $\{(-5,0),(5,0),(0,-5),(0,5)\}$
b. $\{(0,0),(-1,1),(1,1),(-2,4),(2,4),(-3,9),(3,9)\}$
c. $\{(0,0),(1,1),(4,2),(9,3),(16,4),(25,5),(36,6)\}$

Problem 17.7 (Locus from Lines) Determine an equation, or equations, to represent the locus of points equidistant from each pair of lines.

a. $y=x$ and $y=-x$
b. $y=2x+2$ and $y=-2x+2$
c. $y=2x$ and $y=0.5x$

Problem 17.8 (Locus from Radicals) Determine an equation, or equations, to represent the locus of points equidistant from each pair of graphs.

a. $y=-{\sqrt {x}}$ and $y={\sqrt {x}}$
b. $y={\sqrt {x}}+4$ and $y=-{\sqrt {x}}-6$

Problem 17.9 (Flower Bed) The outside edge of a fountain is the locus of points 2 meters from the center. The outside edge of a flower bed is the locus of points 3 meters from the center of the fountain. There is no gap between the fountain and the flower bed. Sketch the flower bed, and find its area.

Problem 17.10 (Describing Loci) Describe and list the locus of points in the plane that are 13 units from the origin, and 12 units from the y-axis.

Problem 17.11 (Three Times the Distance) Find the equation of locus of a point such that its distance from the origin is three times its distance from the x-axis.