The Cartesian form of the hyperbola not centred about the origin, but rather centred at the point (p, q) is similar to the standard form, however we must incorporate the coordinates of the centre point into the equation.
The standard form is:
\frac{x^2}{a^2}-\frac{y^2}{b^2}=1
The equation for a hyperbola centred at (p,q) is given by:
\frac{\left ( x-p \right )^2}{a^2}-\frac{\left (y-q \right )^2}{b^2}=1
The features of this hyperbola remain the same as those of the standard, however, they have been translated by the coordinates (p,q)
- Instead of (±a,0), the vertices are located at (±a+p,q)
- instead of (±ea,0), the foci are located at (±ea+p,q)
- the have equations x = ±(a/e)+p
- asymptotes have the equations: y = q ± (b/a)[x-p]
- the conjugate axis is formed by the coordinates (p,±b+q)
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