In
this section we are going to take a quick look at circles. However, before we
do that we need to
give a quick formula that hopefully you’ll recall seeing at some point in the
past.
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Given
two points (x1, y1) and (x2 , y2) the distance
between them is given by,
So,
why did we remind you of this formula? Well, let’s recall just what a circle
is. A circle is all the points that are the same distance, r – called
the radius, from a point, (h, k
) - called the
center. In other words, if (x, y) is any point
that is on the circle then it has a distance of r from the center, (h, k
).
If
we use the distance formula on these two points we would get,
Or,
if we square both sides we get,
This is the standard form of the equation of a circle with
radius r and center (h, k ).
Do
not square out the two terms on the left. Leaving these terms as they are will
allow us to quickly identify the equation as that of a circle and to quickly
identify the radius and center of the circle.
Graphing
circles is a fairly simple process once we know the radius and center. In order
to graph a circle all we really need is the right most, left most, top most and
bottom most points on the circle. Once we know these it’s easy to sketch in the
circle.
Nicely
enough for us these points are easy to find. Since these are points on the
circle we know that they must be a distance of r from the center.
Therefore the points will have the following coordinates.
right most
point : (h + r , k)
left most point
: (h – r , k)
top most point
: (h , k + r)
bottom most
point : (h , k – r , k)
In
other words all we need to do is add r on to the x coordinate or y
coordinate of the point to get the right most or top most point
respectively and subtract r from the x coordinate or y coordinate
to get the left most or bottom most points.
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Labels:
Mathematician
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