An ellipse with foci \(F_1\) and \(F_2\) is the set of points \(P\) such that \({{PF_1} + {PF_2} = K}\), where \(K\) is a constant such that \(K \gt F_1F_2\). The equation in standard position is \[ \left(\frac{x}a\right)^2+\left(\frac{y}b\right)^2=1 \]
The vertices of the ellipse are \((\pm a, 0)\) and \((0, \pm b)\).
| Focal axis | Foci | Focal vertices | |
| \(a \gt b\) | \(x\)-axis | \((\pm c, 0)\) with \(c=\sqrt{a^2-b^2}\) | \((\pm a,0)\) |
| \(a \lt b\) | \(y\)-axis | \((0,\pm c)\) with \(c=\sqrt{b^2-a^2}\) | \((0,\pm b)\) |
Eccentricity: \({e=\tfrac{c}a}\) \((0\le e \lt 1)\). Directrix: \({x=\tfrac{a}e}\) (if \(a \gt b\)).
A hyperbola with foci \(F_1\) and \(F_2\) is the set of points \(P\) such that \[ {PF_1} - {PF_2} = \pm K \]
where \(K\) is a constant such that \(0 \lt K \lt F_1F_2\). The equation in standard position is \[ \left(\frac{x}a\right)^2-\left(\frac{y}b\right)^2=1 \]
| Focal axis | Foci | Vertices | Asymptotes |
| \(x\)-axis | \((\pm c,0)\) with \(c=\sqrt{a^2+b^2}\) | \((\pm a,0)\) | \(y=\pm\dfrac{b}{a}x\) |
Eccentricity: \({e=\tfrac{c}a}\) \((e \gt 1)\). Directrix: \({x=\tfrac{a}e}\).
A parabola with focus \(F\) and directrix \(\mathcal D\) is the set of points \(P\) such that \({{PF}={P\mathcal D}}\). The equation in standard position is \[ y=\frac1{4c}x^2 \]
Focus \(F=(0,c)\), directrix \(y = - c\), and vertex at the origin \((0,0)\).
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