Key takeaway: A hyperbola is a two-branch curve in which the gap between a point’s distances to the two foci stays fixed, and its branches follow diagonal guide lines.
Grade 12 – PrecalculusG-GPE.A.3Geometry and Measurement
Definition of Hyperbola
A hyperbola is a curve in two separate pieces, called branches, that bend away from each other. Choose two fixed points, the foci. Every point on the curve is farther from one focus than from the other by the same fixed amount.
Far from the center, each branch straightens out and hugs a pair of crossing lines, the asymptotes, without ever touching them. The line through both foci contains the vertices, where each branch turns. Hyperbolas appear in navigation systems, in the sonic boom pattern behind a jet, and in the paths of some comets.
Formula and notation
With center at the origin and branches opening left and right:
\[b^2x^2-a^2y^2=a^2b^2,\qquad c^2=a^2+b^2,\qquad y=\pm\frac{b}{a}x.\]
The vertices are \((\pm a,0)\) and the foci are \((\pm c,0)\). If the minus sign comes before the \(x\) term, the branches open up and down.
Key facts
- The positive fraction in the equation tells which direction the branches open.
- The foci always lie farther from the center than the vertices.
- The asymptotes are the extended diagonals of the box with sides 2a and 2b.
- The constant difference of the focal distances equals 2a.
Hyperbola example
A graph is given by \(9x^2-4y^2=36\). Locate its vertices and foci, and write both asymptote lines.
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Divide every term by 36. The denominators become 4 under \(x^2\) and 9 under \(y^2\), with a minus between them, so \(a=2\), \(b=3\) and the branches open left and right.
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Vertices sit at \(\pm 2\) on the x-axis.
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Focal distance: \(c^2=4+9=13\), so \(c=\sqrt{13}\approx 3.61\).
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Asymptote slopes are \(\pm\frac{b}{a}\), that is \(\pm\frac{3}{2}\).
Vertices at \(\pm 2\), foci at \(\pm\sqrt{13}\), asymptotes \(y=\pm\frac{3}{2}x\).
Common mistake
Watch out
Omar writes the asymptote slopes for this curve as \(\pm\frac{a}{b}=\pm\frac{2}{3}\).
Why it is wrong
For branches opening left and right the slope is \(\frac{b}{a}\), the half-height over the half-width. Test it: a slope of \(\frac{2}{3}\) would make the lines flatter than the branches, which cross them. The right slope is \(\frac{3}{2}\).
Related terms and lessons
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