5.
Circle loci
This
is our problem for this section:
| B is a fixed point and A is any point on circle M. What
is the locus of P, the midpoint of chord AB?
First try to visualise it.
Now check your conjecture by dragging or animating
A in the applet. Then move B and/or M to different positions and
repeat.
Deduce
or prove your conjecture about the locus of P using Synthetic
Geometry.
Click
here for answer: 
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Now
let’s look at the situation from an Analytical Geometry point of
view. In Coordinate Geometry we have to place the information on a coordinate
system, and that raises the problem of generality — we should place
the information to make the mathematics as simple as possible, but be
careful of special cases. Keep this in mind in the following placements
of the information as we seek for a valid proof.
| First,
let’s place the circle at the origin, and let B be any
point on the X-axis as shown in the applet.
Move (drag) or animate point A.
How will you describe the locus of P?
Move B to a different position on
the X-axis. How will you describe the locus of P?
Choose
appropriate coordinates for A, B and P and deduce the equation
and description of the locus of P.
Click
for the solution:
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| Let’s
now look at a more general case: Keep the centre of the circle at M(0, 0), but
place B anywhere on the circle.
Move (drag) or animate point A.
How will you describe the locus of P?
Move B to a different position on
the circle — how will you describe the locus of P?
Choose
appropriate coordinates for A, B and P and deduce the equation
and description of the locus of P.
Click
here for the solution:
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| Let’s
now look at the most general case: Place the circle at any point M on the coordinate system, as shown.
Move (drag) or animate point A. How will you describe
the locus of P? Move M and/or B to different positions — how
will you describe the locus of P?
Choose
appropriate coordinates for A, B, M and P and deduce the equation
and description of the locus of P.
Click
here for the solution:
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Discussion
Which of the above do you see
as valid proofs of the locus of P?
Coordinate Geometry puts us
in a dilemma: To deduce general results, we should beware of
making assumptions through special placements of our information. So we should choose general placements.
However, you should realise
from the above worked examples that as we place our information more generally
on the coordinate system, the algebraic manipulation becomes more complex!
To
keep the manipulation as simple as possible, it is valid to use the simplest
placements without loss of generality, provided we interpret
the special cases generally. For example, it is here not really necessary
to use the most general placement as in Midpoint Locus 3. Our
placement and deduction in the simplest case (Midpoint Locus 1
above) is completely general and valid, provided we do not think of M
as the origin and B as on the X-axis and do not interpret the result 
that the centre of the locus circle lies on the X-axis, which is not generally
true!
So we can place M at the origin and B on the X-axis, provided we think
of it generally, and interpret the result
as the midpoint of the segment MB, without reference to the origin or the
X-axis!
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