1.5 Conic Sections

Big idea

Slice a double cone with a plane and the edge of the cut is a curve whose shape depends only on the tilt of the plane. Cut level and you get a circle; tilt slightly and it stretches into an ellipse; tilt until the plane runs parallel to the side of the cone and it opens into a parabola; tilt further and the plane catches both halves, producing the two branches of a hyperbola. The four curves are one family seen at four angles.

The cone is where the names come from, but it is not how you work with these curves. Each has a description purely in the plane, phrased as a condition on distances. A parabola is the set of points equally far from a fixed point and a fixed line. An ellipse is the set whose distances to two fixed points add to a constant, and a hyperbola the set whose distances to two fixed points differ by a constant. Every standard equation below comes from writing one of those conditions with the distance formula and clearing radicals.

The three definitions look unrelated until you introduce eccentricity. Recast the conditions in terms of one focus and one line, and all three become the same statement: the distance to the focus is a fixed multiple of the distance to the line. The multiple is the eccentricity. Below one gives an ellipse, exactly one a parabola, above one a hyperbola.

That unification pays off in polar coordinates. Put the focus at the pole and the shared definition becomes one short equation with eccentricity as its only shape parameter. This is how conics appear in orbital mechanics, where the attracting body sits at a focus and the eccentricity says whether the motion is a closed ellipse or an unbounded escape.

Decoder

A conic is the set of points whose distance to a fixed focus is a constant multiple of its distance to a fixed line called the directrix, and that constant, the eccentricity, decides which conic appears.

The multiplier is the whole classification. Below $1$ the condition keeps the curve bounded and you get an ellipse. At exactly $1$ the two distances are equal and the curve is a parabola, the borderline case. Above $1$ the curve escapes to infinity in two directions, giving the branches of a hyperbola. A circle is the limit where the eccentricity is zero and the foci have merged; it is the one member of the family with no directrix.

The phrase fixed line is easy to skim past, and the directrix is the piece students most often cannot locate. For a parabola it is as far behind the vertex as the focus is in front. For an ellipse or hyperbola there are two directrices, one per focus, at distance $a/e$ from the centre and always further out than that focus.

The usual mistake with the standard equations is mixing up which letter is which. In every standard form $a$ belongs to the vertices, so $a^2$ sits under the variable whose axis the curve stretches along. For an ellipse $c^2 = a^2 - b^2$ and the foci are inside; for a hyperbola $c^2 = a^2 + b^2$ and the foci are outside the vertices.

Definitions and results

Parabola. The set of points equidistant from a focus and a directrix. With focus $(p,0)$ and directrix $x = -p$ the equation is $y^2 = 4px$; with focus $(0,p)$ and directrix $y = -p$ it is $x^2 = 4py$. Each opens toward the focus, the vertex sits halfway between focus and directrix, and the eccentricity is $1$.

Ellipse. The set of points whose distances to two foci sum to $2a$. In standard position with a horizontal major axis,

$$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \qquad a > b > 0, \qquad c^2 = a^2 - b^2 $$

with vertices $(\pm a, 0)$, co-vertices $(0, \pm b)$ and foci $(\pm c, 0)$. If the larger denominator sits under $y^2$, the major axis is vertical instead.

Hyperbola. The set of points whose distances to two foci differ in absolute value by $2a$. In standard position opening left and right,

$$ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1, \qquad c^2 = a^2 + b^2 $$

with vertices $(\pm a, 0)$, foci $(\pm c, 0)$, and asymptotes $y = \pm \frac{b}{a}x$. The branches open along the axis of the positive term.

Asymptotes as a construction. Draw the rectangle with corners $(\pm a, \pm b)$; its extended diagonals are the asymptotes, which the hyperbola hugs without ever meeting.

Translation. Moving the centre or vertex to $(h,k)$ replaces $x$ by $x-h$ and $y$ by $y-k$. An equation with linear terms in $x$ or $y$ is a translated conic, and completing the square recovers the standard form.

Eccentricity. Define $e = c/a$ for an ellipse or hyperbola and $e = 1$ for a parabola, so $0 \le e < 1$ for ellipses, with $e = 0$ the circle, and $e > 1$ for hyperbolas. It measures elongation, not size: two ellipses with the same eccentricity are scaled copies.

Directrices. In standard horizontal position they are $x = \pm a/e$, each paired with the focus on its own side, and the focus-directrix ratio equals $e$ at every point of the curve.

Polar equation of a conic. With a focus at the pole and the directrix the vertical line $x = p$ for $p > 0$,

$$ r = \frac{ep}{1 + e\cos\theta} $$

A minus in place of the plus puts the directrix at $x = -p$, and sine in place of cosine puts it at $y = \pm p$. To read a given equation, divide until the denominator’s constant term is $1$: the trigonometric coefficient is $e$ and the numerator is $ep$.

Worked examples

A parabola from its focus and directrix

Find the equation of the parabola with focus $(0,2)$ and directrix $y = -2$.

Apply the definition directly. A point $(x,y)$ is equidistant from the focus and the line, and the distance to a horizontal line is the vertical gap:

$$ \sqrt{x^2 + (y-2)^2} = |y+2| $$

Square and expand: $x^2 + y^2 - 4y + 4 = y^2 + 4y + 4$. The $y^2$ and constant terms cancel, leaving $x^2 = 8y$.

Check against the standard form $x^2 = 4py$: here $4p = 8$, so $p = 2$, which puts the focus at $(0,2)$ and the directrix at $y = -2$, as given. The vertex is the origin, halfway between them.

Reading an ellipse off its equation

Describe the curve $9x^2 + 25y^2 = 225$.

Divide through by $225$ to get the constant term to $1$:

$$ \frac{x^2}{25} + \frac{y^2}{9} = 1 $$

The larger denominator is under $x^2$, so the major axis is horizontal with $a = 5$ and $b = 3$, and $c^2 = 25 - 9 = 16$ gives $c = 4$. The vertices are $(\pm 5, 0)$, the co-vertices $(0,\pm 3)$, the foci $(\pm 4, 0)$, and $e = 4/5$.

Check the defining property at the co-vertex $(0,3)$. Its distance to each focus is $\sqrt{16+9} = 5$, so the sum is $10 = 2a$, as required. The directrices are $x = \pm a/e = \pm 25/4$.

A hyperbola and its asymptotes

Describe the curve $16x^2 - 9y^2 = 144$. Divide by $144$:

$$ \frac{x^2}{9} - \frac{y^2}{16} = 1 $$

The positive term carries $x$, so the branches open left and right, with $a = 3$ and $b = 4$. For a hyperbola the relation adds, so $c^2 = 9 + 16 = 25$ and $c = 5$. The vertices are $(\pm 3, 0)$, the foci $(\pm 5, 0)$, the asymptotes $y = \pm \frac43 x$, and $e = 5/3 > 1$.

Check the focal difference at the vertex $(3,0)$: its distances to the foci are $2$ and $8$, differing by $6 = 2a$. The numbers $3,4,5$ appeared in the ellipse above in different roles: there $5$ was a vertex distance, here a focal distance.

A conic given in polar form

Identify and describe $r = \dfrac{4}{2 + \cos\theta}$.

The standard form needs a bare $1$ leading the denominator, so divide through by $2$:

$$ r = \frac{2}{1 + \tfrac12 \cos\theta} $$

Read off $e = \frac12$, below $1$, so this is an ellipse with a focus at the pole. The numerator is $ep = 2$, so $p = 4$ and the plus sign puts the directrix at $x = 4$.

Locate the vertices along the major axis. At $\theta = 0$, $r = \frac43$, giving $\left( \frac43, 0 \right)$; at $\theta = \pi$, $r = 4$, giving $(-4,0)$. The major axis runs from $-4$ to $\frac43$, so $2a = \frac{16}{3}$, $a = \frac83$, and the centre is the midpoint $\left( -\frac43, 0 \right)$.

Check for consistency: the distance from the centre to the focus at the origin is $\frac43$, so $c = \frac43$, and $c/a = \frac12$, the eccentricity read off at the start.

Practice

Begin with parabolas, where one focus and one line are involved. Locate the vertex halfway between them and read the opening direction from the sign.

Next ellipses. Get the equation into the form with $1$ on the right, see which denominator is larger, and use $c^2 = a^2 - b^2$ before reporting foci or eccentricity.

Last hyperbolas, with two changes that matter: the relation becomes $c^2 = a^2 + b^2$, and the asymptotes come from the rectangle built on $a$ and $b$.

Quiz

Six items across parabolas, ellipses, hyperbolas, eccentricity and the polar form.