Introduction to Ellipses
The classic example of constructing an ellipse is to place two pins some distance apart. Tie each end of a string around each pin and then pull the string taut with a pencil. Then trace the curve that the string will allow, keeping it taut.
Try this in the applet below. The two pins are located at $F$F and $F'$F′. The pencil is located at the point $P$P.
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The shape created in this way is always an ellipse. The points located at the pins are the foci (or focus if singular).
Notice that the length of the string never changes. So the length of string from one focus to a point on the ellipse added to the length of string from the other focus to the same point will always equal a constant for that ellipse. The constant for the above ellipse is $4$4.
The distance from one foci to a point on the ellipse added to the distance from the other foci to the same point is equal to a constant for the same ellipse.
$FP+F’P=\text{constant}$FP+F’P=constant
Worked example
Consider the ellipse below.
Find the value of $x$x.
Think: A pair of lines meeting at a point on the ellipse from the foci will have the same total length. There are two pairs of these lines on this ellipse. The sum of one pair must be equal to the sum of the other pair.
Do: Construct an equation relating the length of the two pairs of lines and solve for $x$x.
| $GP+FP$GP+FP | $=$= | $GQ+FQ$GQ+FQ | |
| $x+x$x+x | $=$= | $9+3$9+3 | (Substitution) |
| $2x$2x | $=$= | $12$12 | (Collecting like terms) |
| $x$x | $=$= | $6$6 | (Dividing both sides by $2$2) |
The following applet allows you to investigate the string and pin construction of an ellipse. You can vary the distance between the pins (foci) and change the length of the string. You can trace out the path yourself by dragging the point around (make sure the Trace button is ON), or let the animation button trace it for you.
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Practice questions
Question 1
Consider the following ellipse. The points given are the foci of the ellipse. Find the value of $x$x.
question 2
Mario picks out a piece of string $8$8 inches long. He fixes each end of the string at points $F$F and $G$G on a chalkboard as shown, such that $F$F and $G$G are $6$6 inches apart and equidistant to point $O$O. He pulls the string taut and then uses chalk to trace out the curve shown below.
What is the length of $\overline{OF}$OF?
What is the length of $\overline{FR}$FR?
What is the length of $\overline{OR}$OR?
What is the length of $\overline{FP}$FP?
What is the length of $\overline{OP}$OP?