Find the vertices, foci, asymptotes, and eccentricity of any hyperbola from its standard equation.
Ellipse: c² = a² − b², e < 1
Hyperbola: c² = a² + b², e > 1
The key difference is the sign in the c² formula. For an ellipse, c² = a² − b² (b² is subtracted, so c < a). For a hyperbola, c² = a² + b² (b² is added, so c > a).
Eccentricity: an ellipse always has e < 1 (foci inside the curve), while a hyperbola always has e > 1 (foci outside the curve, beyond the vertices).
Shape: an ellipse is a single closed curve; a hyperbola has two separate open branches that curve away from the center in opposite directions.
y − k = ±(b/a)(x − h) [horizontal]
y − k = ±(a/b)(x − h) [vertical]
Draw a rectangle centered at (h, k) with width 2a and height 2b. The diagonals of this rectangle are exactly the asymptotes of the hyperbola.
The hyperbola's branches approach the asymptotes but never touch or cross them as x → ±∞. The asymptotes give the hyperbola its characteristic flared shape.
Notice the slopes swap between horizontal and vertical: horizontal uses ±b/a (rise over run where b is height, a is half-width), vertical uses ±a/b.
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