Difference of Distances

Algebra 2 · Axiom Academy

LESSON Difference of Distances A hyperbola is every point whose two distances to two fixed foci keep a constant difference, . Fix two points called the foci , F_1 and F_2 , a distance 2c apart on an axis. For any point P in the plane there are two distances that matter: d_1 , from P to F_1 , and d_2 , from P to F_2 . The two fixed foci, a distance 2c apart The two distances from a moving point P 2. The Difference Stays Constant A hyperbola is the set of all points P for which the absolute difference of the two distances is a fixed constant. We call that constant 2a . Watch the two segments below. Both d_1 and d_2 grow and shrink as P slides, yet is pinned at 2a = 6 the whole time. Because the difference can be taken either way, the locus splits into two branches , one hugging each focus. Points closer to F_2 : here d_1 - d_2 = +2a . Points closer to F_1 : here d_2 - d_1 = +2a . 3. Sum vs. Difference — and the Equation The same two foci build both classic conics. Fix the sum of the distances and you trace a closed ellipse ; fix the difference and you trace an open hyperbola . One closed curve surrounding both foci. Two open branches bending away from each other. Take a = 3 and b = 4 . Then . The vertices are at and the foci at , and every point on the curve satisfies . A hyperbola comes down to one idea: the difference of the distances to two foci stays constant.

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