7.
More loci
Steps in finding
a locus:
- Decide what information
is given or fixed and what is variable.
- Locate one
of the points of the locus, and label it with it’s coordinates
as variables: e.g. P(x,
y).
- Draw/locate as
many points as necessary to give you a "picture"
of the locus.
- Formulate relationships to find the algebraic equation of the locus, and describe it geometrically.
Let’s
start with a familiar example:
Find the locus of a set of points that are equidistant from
two points A and B.
You know that
geometrically, the answer is the perpendicular bisector
of AB as illustrated in the applet. We now want to look at the
problem from the perspective of Analytical Geometry.
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Use Analytical Geometry to find the
locus of points equidistant from the point A(4, 2) and B(6, 8).
Click here for the Solution:  |
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Open the Line Applet: 
What is
the locus of the midpoint P of a line segment OM, if O is
fixed, while M moves along a line?
Our investigation will move from special cases to the general:
- The
applet opens with a special O at the origin,
and M on the special horizontal line y
= b.
Intuitively visualise the location (locus) of all points P as M moves along the line …
Drag or animate point M to check your conjecture ...
Find the equation of the locus of P algebraically. Describe the locus geometrically
- What if O is not at the origin?
Drag O to a new position and clear traces. Drag or animate point M ...
Find the equation of the locus of P algebraically. Describe the locus geometrically
- What if line QC is not horizontal?
Drag C or Q to a new position and clear traces. Drag or animate point M ...
Find the equation of the locus of P algebraically. Describe the locus geometrically
Click here for
a discussion:  |
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Open the Ordinate Applet: 
What is the
locus of the midpoint P of a line segment MN, with M any point
on any line and MN perpendicular to the X-axis? Our investigation will move from special cases to the general:
- Locus 1 is a special line through the origin, and
a special N on the X-axis.
Intuitively visualise the shape
of the locus …
Find the equation of the locus of P.
Animate to check your results ...
- Now
take a more general case: What if the line does not pass through the origin?
Click “Locus 2”
and clear the traces.
What is now the equation of locus P? Describe the locus.
- Now
take a more general case: What if N is not on the X-axis, but on any horizontal line?
Click Locus 3
and clear traces. What is now the equation of locus P? Describe the locus.
- Now
take a more general case: What if N is not on a horizontal line, but on any line?
Click Locus 4
and clear traces. What is now the equation of locus P? Describe the locus.
Click here for
a discussion:
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