Identify the constant difference for a hyperbola with foci (−6, 0) and (6, 0) and a point on the hyperbola (4, 0).

Answer 1

Very good Yefim, agree, job well done. Since I was working on it but the same time you were, I will offer the same rationale. I will try to answer this one without the benefit of a visual. We can at least use the characteristics of a hyperbola. In this case with the foci at (6, 0) and (-6, 0) and the vertex would be where the hyperbola crosses the X axis at (4, 0) and (-4, 0). This means that a = 4, (from (0, 0) to the Vertex). We also know that the distance (D2) from any point (X, Y) on the hyperbola to the furthest away foci - distance (D1) from the point (X, Y) to the nearest foci is a constant. So in this case we have:

D2 = 4+6, D1 = 2, we can now say that the absolute value of D2 - D1 equals 8.

This also validates a check that the absolute value of D2 - D1 = 2a = 2 time 4 = 8

Answer 2

Real axis is x-axis. Equation of hHyperbola: x²/a² - y²/b² = 1. From condition we c = 6. Because a² + b² = 6²,

a² + b² = 36. And point (4,0) on hyperbola we have 4²/a² - 0 = 1, a² = 16 and a = 4. Constant difference is 2a = 8. Now b² = 36 - 16 = 20 and equation of hyperbola: x²/16 - y²/20 = 1