3.
Circle not at the origin
Again: What is the
locus of points equidistant from a fixed point?
In
the previous section we placed the fixed point at the origin. This is
of course an assumption — the resulting equation is therefore a
special case which only applies in the special case when the fixed point
is at the origin. But what if the
fixed point is not at the origin?
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Open the Moving Circle applet: 
The
applet shows the graph of a circle as the locus of all points
a distance r from a fixed point M(a, b) and the equation of
the circle for specific values of a, b and r.
Move
point M (drag it, or click the sliders), and watch the equation change ....
Develop a theory connecting the values of a, b and r, the equation and the graph of the circle.
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We
use different approaches to find the equation of the circle in the following
two activities.
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Open the Translated Circle applet below.
In
the applet, click the Shift buttons and carefully observe how
the coordinates of P, P' and P'' change.
Animate or move points P and (a, b) and observe how the coordinates
change.
How can you use this information to deduce the equation of the
circle with centre (a, b) and radius r?
 
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Use the definition of the
distance between two points to find the locus of points
that are equidistant from a fixed point.
Click
for a discussion:
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The standard form of the equation of the
circle with centre at point (a, b) and radius r
is:
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In the applet, animate or drag point A.
Clear traces and then animate point B.
Find the equations generating the two loci.
Click
for a discussion:
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