The standard equation of a hyperbola describes one that is centred about the origin, symmetrical about the y-axis and opens from the vertices in an east-west direction. So what about hyperbolas that open in the north-south direction?
Well, the equation of north-south hyperbolas is very similar to the standard - just the variables x and y are swapped around. Hence we have...
\frac{y^2}{a^2}-\frac{x^2}{b^2}=1
Now likewise, the features are also reversed.
- Vertices: (0,±a)
- Focal points: (0,±ae)
- Directrices: y = ±a/e
- Asymptotes: y = ±(a/b)x
The relationship (ae)2 = b2 + a2 still holds true.
In this video, we also sketch the equation y2 - 4x2 = 16 using these features and relationships. In order to do this, we need to modify this equation into the north-south form by first dividing both sides by 16,
From this we get...\frac{y^2-4x^2}{16}=1
\frac{y^2}{4^2}-\frac{x^2}{2^2}=1
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