Thursday
Thursday: A Constant Difference, the Hyperbola
Wren holds up his clipboard. Two curves bend away from each other, like a pair of parentheses.
"Same two stakes," he says. "But now the distances differ by a fixed amount instead of adding to one."
"Distance to the far stake minus distance to the near stake equals 6," Comet reads. "What can we make of that?"
Nova projects two crossing dashed lines through the center. "Would you like a hint? Far out, the curves hug these lines."
"Asymptotes," Wren says. "And the vertices sit at ±3, half the difference. Just like a was half the sum."
"Then c² must be a² plus b² this time," Comet says. "The stakes are outside the curve, at ±5."
"Same algebra, one sign flipped," Wren says. "Let us run it."
The definition of a hyperbola
A hyperbola is the set of points whose distances to two foci differ by a constant. Call the constant 2a.
The curve has two branches, one near each focus. Each branch crosses the axis at a vertex, (±a, 0).
The foci sit outside the vertices, at (±c, 0) with c greater than a. Set b² = c² - a², so c² = a² + b².
The derivation, read first
- Foci at (-c, 0) and (c, 0). Let P = (x, y) be a point whose focus distances differ by 2a.
- √((x + c)² + y²) - √((x - c)² + y²) = ±2a. The sign depends on which branch P is on.
- Move one root across, square both sides. One root remains.
- Isolate it and square again. Every root is gone.
- Collect terms: (c² - a²)x² - a²y² = a²(c² - a²).
- Let b² = c² - a² and divide by a²b²: x²/a² - y²/b² = 1.
Compare with the ellipse. The sum became a difference, so a² - c² became c² - a², and the plus in the equation became a minus.
For the clipboard curve: a = 3, b = 4, so c² = 9 + 16 = 25 and c = 5. The equation is x²/9 - y²/16 = 1.
The asymptotes
Far from the center, the 1 on the right side hardly matters. Replace it with 0: x²/a² - y²/b² = 0.
Solve for y: y = ±(b/a)x. These two lines are the asymptotes. The branches get closer to them and never cross.
For the clipboard curve, y = ±4/3 x. Draw a box 6 wide and 8 tall at the center; its diagonals are the asymptotes.
- ?Foci at (±c, 0) and a point P whose focus distances differ by 2a
- ?Write the difference of two square roots equal to ±2a
- ?Move one root across and square; one root remains
- ?Let b² = c² - a² and divide: x²/a² - y²/b² = 1
- ?Isolate that root and square again; the roots are gone
- ?Collect: (c² - a²)x² - a²y² = a²(c² - a²)
A mixed set
- Read the question.
- Tap your answer.
The clipboard hyperbola has vertices at (±3, 0) and b = 4. What is its equation?Where are the foci of the hyperbola x²/9 - y²/16 = 1?What are the asymptotes of the hyperbola x²/9 - y²/16 = 1?
What are the asymptotes of the hyperbola x²/4 - y²/36 = 1?Which answer makes sense?
An ellipse with a = 5 and b = 3 has c = 4, less than a: the foci are inside. A hyperbola with a = 3 and b = 4 has c = 5, more than a: the foci are outside.
If a computed c for an ellipse comes out bigger than a, the formula was the hyperbola's. Check the sign in the equation first.
For a hyperbola, the difference 2a must be less than the focus gap 2c, or no point can exist.
| Statement | True or false? |
|---|---|
| A hyperbola has two branches. | ? |
| For a hyperbola, c² = a² + b². | ? |
| The branches eventually cross their asymptotes. | ? |
| 9 + 16 = 25 | ? |
| The foci of a hyperbola sit between its vertices. | ? |
Clear reasoning. Tomorrow all four conics line up, including the circle and parabola from Geometry.