Consider a hyperbola and find its foci. Consider some points on the hyperbola and find the distances from them to the foci. Is the sum of the distances constant? What about the difference?
False. 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.
Practice makes perfect
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 F1(2.646,0)
Distance to F2(-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.
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