Moving a Circle in a Parabola

ice cream

steepness k

1

y coordinate c of center

4.25

answer

This Demonstration shows that the center

B

of a unit circle tangent to the parabola

y=

2

x

is at (0, 5/4). In addition, the segment connecting the point

B

and the tangent point

A

makes a 60° angle with the vertical axes of the parabola.

The radius of the tangent circle with center on the

y

axis at

(0,c)

is

r=

(4ck-1)

(2k)

, where

k

is the steepness of the parabola

y=k

2

x

.

Details

Checking the "answer" checkbox gives the radius of the purple circle. When the circles are tangent, the values of the radii are 1, 2, 3, 4, ….

Checking the "ice cream" checkbox shows the three tangent spheres and the ratio of their volumes to the volume of the paraboloid obtained by revolving

y=

2

x

about the

y

axis.

References

[1] J. Stewart, Calculus: Early Transcendentals, 5th ed., Belmont, CA: Brooks/Cole, 2007, Chapter 3.

External Links

Circle (Wolfram MathWorld)

Parabola (Wolfram MathWorld)

Paraboloid (Wolfram MathWorld)

Permanent Citation

Abraham Gadalla

"Moving a Circle in a Parabola"
http://demonstrations.wolfram.com/MovingACircleInAParabola/
Wolfram Demonstrations Project
Published: July 18, 2011