Let's begin drawing a hyperbola with its foci. Then we will consider some points on the hyperbola and will find their distances to the foci.

In the table below we summarize the distances between the points

P,

Q,

and

R

from the foci. We also find the sum of the distances and the absolute value of the difference.

Point	Distance to $F_{1} (2.646, 0)$	Distance to $F_{2} (- 2.646, 0)$	Sum of Distances	Difference of Distances
$P (3, 1.936)$	$1.969$	$5.969$	$∣ 1.969 + 5.9689 ∣ = 7.938$	$∣ 1.969 - 5.969 ∣ = 4$
$Q (- 3.3, - 2.273)$	$6.365$	$2.365$	$∣ 6.365 + 2.365 ∣ = 8.73$	$∣ 6.365 - 2.365 ∣ = 4$
$R (2.5, - 1.299)$	$1.307$	$5.307$	$∣ 1.307 + 5.307 ∣ = 6.614$	$∣ 1.307 - 5.307 ∣ = 4$

From the table above we see that the sum of the distances is not constant, which means that the given statement is false. However, we notice that the absolute value of the difference of the distances is constant.

A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to two given points is constant.

Exercise

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