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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.

Case1step1.JPG

 

 

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

 

Case1step2.JPG

 

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

 

Case1step3.JPG

 

We construct a line that passes through OD and marks point C. Then, we connect points OC 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.

 

Case1step4.JPG

 

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

Case1step5.JPG

 

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.

 

Case1step6.JPG

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.

 

case2.JPG

 

 

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 ellipses 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+OF is constant as well

 

 

case1ellipse.JPG

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.

 

case2ellipse.JPG

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-OF| is a constant.

 

case1hyperbola.JPG

 

 

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.

 

Case2Hyperbolas.JPG

 

 

 

 

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