Conic sections from equations
The ellipse, parabola and hyperbola: from the canonical equation find the semi-axes, the foci, distances to a focus — and tell which curve an equation describes.
How to solve it
An ellipse is the set of points whose distances to two foci add up to a constant; a hyperbola, where the difference is constant; a parabola, the points equally far from a focus and a line, the directrix. Everything needed is read straight off the canonical equation.
Step by step
- Recognize the curve by the signs of the squares: both plus with the sum equal to $1$ is an ellipse, different signs a hyperbola, only one variable squared a parabola.
- The ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $a > b$: the foci are $(\pm c,\ 0)$ with $c^2 = a^2 - b^2$.
- The hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$: $c^2 = a^2 + b^2$, and the difference of the distances to the foci is $2a$.
- The parabola $y^2 = 2px$: the focus $\left(\frac{p}{2},\ 0\right)$, the directrix $x = -\frac{p}{2}$; the distance from a point to the focus is $x + \frac{p}{2}$.
Common mistakes
- Adding the squares for an ellipse as for a hyperbola: for an ellipse $c^2 = a^2 - b^2$, for a hyperbola it is plus.
- Using $a^2$ instead of $a$: the equation has squares, the semi-axis is the root of the denominator.
- Mixing up $2p$ and $p$ in $y^2 = 12x$: $2p = 12$, $p = 6$, the focus is at $(3,\ 0)$.
- Not bringing the equation to canonical form: divide $16x^2 + 9y^2 = 144$ by $144$ first.