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Chapter 4: Problem 52

Use two thumbtacks, a loop of string, and a pencil to draw several ellipses. Describe how the shape of an ellipse varies as the distance between the thumbtacks changes.

Short Answer

Expert verified
Drawing ellipse with thumbtacks, a loop of string, and a pencil forms a circular shape when the thumbtacks (foci) are close and a more elongated or oval shape when they're further apart.

Step by step solution

01

Materials Preparation

Gather two thumbtacks, a loop of string, and a pencil. Ensure the string is long enough to stretch comfortably between two thumbtacks and around the pencil.
02

Initial Ellipse Drawing

Place the thumbtacks in a piece of cardstock or corkboard, at any distance apart. Loop the string around the thumbtacks and the pencil, and ensure the pencil can move comfortably without the string being slack. Draw an ellipse by moving the pencil around, keeping the string taut at all times.
03

Adjust Distance and Draw More Ellipses

Move the thumbtacks closer or further apart and repeat the process of drawing an ellipse with the string and pencil. Observe and note the changes in the shape of the ellipse.
04

Analyze the Shapes

Look closely at the ellipses drawn and compare them. As the thumbtacks (focal points) move closer together, the ellipse becomes more circular. As the thumbtacks move further apart, the ellipse becomes more elongated and narrow. This is due to the constant sum of distances - when thumbtacks are closer together, there is less string length to 'distribute' along the curve, making it rounder.

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