The equation of a circle is
(x - h)² + (y - k)² = r²
We will need the center(h,k) and the radius r. Let's draw the
graph:
We need the center and the radius. Since the circle is tangent
to the y-axis at (0,6), we know that a radius drawn at that point
will be horizontal. So the y-coordinate, k, of the center is the
same as the y-coordinate of the point k = 6, so the center is
(h,6). We don't know h, and must find it. However we know that
h = r. So the center (h,6) is really (r,6) So we draw the radius
(in green) from (0,6) to the center (r,6):
From the center (r,6) we draw a perpendicular to the x-axis. It is
6 units long because the y-coordinate of the center is 6. Also it
bisects the 16-unit chord, dividing it into two 8-unit segments:
Next we draw a radius from the center (r,6) to the left end of the chord:
Now we have a right triangle and we can use the Pytagorean theorem
to find r:
r² = 8² + 6²
r² = 64 + 36
r² = 100
r = 10
So the figure is now:
And the equation of the circle,
(x - h)² + (y - k)² = r²
becomes,
(x - 10)² + (y - 6)² = 10²
or
(x - 10)² + (y - 6)² = 100
There is another possible solution. That's because we could
have drawn the circle tangent on the left side of the y-axis,
and the center would have been (-10,6) and the equation would
have been
(x + 10)² + (y - 6)² = 100
And the graph would be the exact mirror image of the one above.
Edwin
