This document discusses loci and how to write equations representing loci given certain conditions. It begins by defining a locus as a set of points traced by a moving point based on a specific rule or condition. Examples are then given of writing equations for loci of points equidistant from two given points, finding the midpoint, or a given distance from one point. The key is setting up an equation relating the distances defined by the condition and solving for the relationship between x and y.

1. LOCUS
Writing equations based on
descriptions.

2. What is a locus?
 Imagine a point on the Cartesian plane that is in
motion…
It can move along a line...
It can make a lot of turns...
It can follow a circular path...

3. What is a locus?
 Since the path consists of different coordinates, a
locus can also be defined as a set of points, and
only the points that satisfy a specific condition.
 A locus is the path that is traced by the moving
point and the movement of the point is based on
a specific rule or condition.
 Because a locus is a set of points, it can be
represented by an equation.
 And that’s what this set of slides is about. We will
write equations for a locus given a condition.

4. Equation of a Locus?
1. What is the equation that represents a locus of
points equidistant from (-3, -2) and (-3, 4)?
The concept involved here is distance between points.
The condition means that the any random point in the locus we
are looking for has the same distance from (-3, -2) and (-3, 4).
Since we already know how to represent distance between two
points and by letting (x, y) be any point in the locus, we have…
22
23 yx as the distance of (x, y) from (-3, -2)
22
43 yx as the distance of (x, y) from (-3, 4)

5. 1. What is the equation that represents a locus of
points equidistant from (-3, -2) and (-3, 4)?
2222
4323 yxyx
The condition says that the distances are the same. So…
We only need to simplify this to come up with the equation
for the locus following the description.

6. 1. What is the equation that represents a locus of
points equidistant from (-3, -2) and (-3, 4)?
Square both sides of
the equation.
(x + 3)2 can be
cancelled out since
it’s on both sides of
the equation.
Combine like terms
and simplify.

7. Thus, the equation that represents a locus of points
equidistant from (-3, -2) and (-3, 4) is given by
But, is there another way to find the equation without
having to solve for the equation?
In this case, there is, if we look at it graphically.
1y

8. Here are the points when plotted.
Since we are looking for points that are
equidistant to both of these points, the
first point which is equidistant to these
two is their midpoint found at (-3, 1).
midpoint!
And all the points that are equidistant to
(-3, 4) and (-3, -2) are located along the
line y = 1.
y = 1
All we had to do was compute for the midpoint of (-3, 4)
and (-3, -2) and take the ordinate!!

9. 2. What is the equation that represents a locus of
points equidistant from (5, -5) and (-1, -5)?
Let’s visualize this first…
Let’s try this…
The midpoint is at (2, -5).
But unlike the 1st locus, the
locus here is a vertical line.
Which means that the equation
is represented by x = 2.
midpoint!
x = 2
But is the equation of a locus always a
horizontal line or a vertical line?

10. Of course not! Just look back at the 2nd slide.
A locus can be any curve.
But at least for loci that
are vertical and horizontal
lines, we have a shorter way
of finding out its equation.

11. 3. What is the equation that represents a locus of
points equidistant from (-2, 5) and (3, -3)?
Let’s visualize this first…
Let’s try this…
Obviously, we can’t have a
vertical or a horizontal line to
represent this locus. But it
should contain the midpoint.
To get the equation of this
line, let us solve it the way we
did the first example.
midpoint!

12. 2222
3352 yxyx
The condition says that the distances are the same. So…
Simplifying this equation will give us…
3. What is the equation that represents a locus of
points equidistant from (-2, 5) and (3, -3)?

13. The equation of that loci that we will focus on are
those that will follow the form
So, the examples that we had should be written as:
Example 1:
Example 2:
Example 3:
Let us try some other examples and write the equations in
the form shown above.
022
EDyCyBxAx
01y
02x
0111610 yx

14. 2222
312 yyxyx
Now, before we jump to an equation, notice that this is
different from the past conditions.
Simplifying this equation will give us…
4. What is the equation that represents a locus of
points equidistant from (-2, 1) and x = 3?
This condition involves a point and a line. The line x = 3 is
composed of infinitely many points that look like (3, y).
The set-up to get the equation is then given by…

15. 4. What is the equation that represents a locus of
points equidistant from (-2, 1) and x = 3?
2222
0312 xyx

16. 5. What is the equation that represents a locus of
points whose distance from (3, 2) is fixed at 5 units?
523 22
yx
This condition talks about a fixed distance. Which means that the
locus is a set of points all of which is 5 units away from (3, 2).
The set-up to get the equation is then given by…

17. 6. What is the equation of the locus consisting of
points whose distance from (-2, 2) is two-thirds of the
distance from (6, -6)?
2222
66
3
2
22 yxyx
The set-up to get the equation is then given by…

18. This ends the slides on
equation of a locus.
Off to the next topic…