Mathematics Education

EMAT 6680, Professor Wilson

Exploration Number 7, Tangent Circles by Ursula Kirk

Case 1

We start the exploration with two circles, one with a center at O and the other with a center to O’ and a point on one of the circles.

We choose point D on the biggest circle as our point of tangency.

We construct a circle with a center at D and the same radius that circle O’.

We construct a line that passes through OD and marks point C. Then, we connect points O’C with a segment, and we find the point of this segment at E. From midpoint E we construct a segment bisector. The segment bisector will intercept OC at F.

Joining O’ and F with a line segment will help us construct a triangle CFO’

With a center at F and a radius FD, we can construct a circle that is tangent to both circle O and circle O’, our original circles.

Case 2

Similar to case 1, we can construct a second case, case 2. In the second case, circle H is also tangent to circle O’ and circle O.

Considering these two cases, we can form ellipses and hyperbolas by tracing points F and H

Ellipses

Case 1

In geometry an ellipse is the locus traced by a point moving in a plane so that the sum of its distances from two other points called foci is a constant. The foci are located on the ellipse’s major axis. Our foci for case 1 are at O’ and O. The ellipse is found by tracing the locus of point F.

Since the sum of the distances from any point of the ellipse to O and O’ is constant and equal to the major diameter, then OF+O’F is constant as well

Case 2

Similarly, for case 2, we can also find our ellipse by tracing the locus of point H. In this case my foci are also points O and O’.

Hyperbolas

In geometry a hyperbola is the locus of a point where the absolute value of the difference of the distance to the foci is constant.

Case 1

In case 1, the locus of the hyperbola is formed by tracing point F and the foci are at O and O’ which are the centers of our original circles. Therefore |OF-O’F| is a constant.

Case 2

Similarly, for case 2, we find the locus of the hyperbola by tracing point H and the foci of our hyperbola are O and O’.