🔮 Conic Sections
⚡

Hyperbolas

Identify the equation, foci, vertices, and asymptotes of a hyperbola.

A hyperbola is the set of all points where the absolute difference of distances from two fixed foci is constant. Unlike an ellipse, a hyperbola has two separate branches and two asymptotes.

Horizontal Hyperbola

(x−h)²/a² − (y−k)²/b² = 1Opens left and rightVertices: (h±a, k)Asymptotes: y − k = ±(b/a)(x − h)

Vertical Hyperbola

(y−k)²/a² − (x−h)²/b² = 1Opens up and downVertices: (h, k±a)Asymptotes: y − k = ±(a/b)(x − h)
c2=a2+b2(hyperbola — c is larger than both a and b)c^2 = a^2 + b^2 \qquad \text{(hyperbola — c is larger than both a and b)}
✏️Analyze x²/9 − y²/16 = 1
Horizontal hyperbola (x² positive). a² = 9, b² = 16, so a = 3, b = 4. Vertices: (±3, 0). c = √(9+16) = 5. Foci: (±5, 0). Asymptotes: y = ±(4/3)x.
⚠️

Key difference from ellipses: for hyperbolas, c² = a² + b² (c is largest). For ellipses, c² = a² − b² (c is smallest). Watch the sign in the equation.

✏️ Try It!

For the hyperbola y²/4 − x²/9 = 1, which direction do the branches open?

Take Quiz 📝 — 25 Questions