09 Conic Sections

Recognizing the Three Conics

A conic section is a curve formed by intersecting a plane with a double cone. In analytic geometry, conics are represented by second-degree equations involving $x$ and $y$, but each principal conic also has a geometric distance definition.

• A Parabola has equal distances from a focus and a directrix.

• An Ellipse has a constant sum of distances from two foci.

• A Hyperbola has a constant absolute difference of distances from two foci.

For a nonrotated equation, inspect the squared terms first:

• Only one variable is squared: parabola.

• Both variables are squared with the same sign but different coefficients: ellipse.

• The squared terms have opposite signs: hyperbola.

• The squared terms have equal positive coefficients: circle, which is a special ellipse.

These quick classifications apply to equations without an $xy$ term. An equation containing $xy$ may describe a rotated conic and generally requires a rotation-of-axes method or technology for a complete graph.

Takeaway: Identify the conic from the squared terms, then convert to standard form to reveal its graphing features.

Parabolas: Vertex, Focus, and Directrix

A parabola is identified by its vertex, axis of symmetry, opening direction, focus, and directrix. Its standard forms are

for a horizontal parabola and

for a vertical parabola.

The vertex is $(h,k)$. For a horizontal parabola, the focus is $(h+p,k)$, the directrix is $x=h-p$, and the axis of symmetry is $y=k$. For a vertical parabola, the focus is $(h,k+p)$, the directrix is $y=k-p$, and the axis of symmetry is $x=h$. The sign of $p$ gives the opening direction.

To graph a parabola:

1. Identify the vertex $(h,k)$.

2. Determine whether the equation is horizontal or vertical.

3. Use the sign of $p$ to determine the opening direction.

4. Plot the focus and directrix when useful.

5. Plot symmetric points on both sides of the axis.

For example, consider

Because $8=4p$, $p=2$. Thus the vertex is $(2,-1)$, the parabola opens upward, the focus is $(2,1)$, and the directrix is $y=-3$. Setting $x-2=\pm2$ gives the symmetric points $(0,-\frac{1}{2})$ and $(4,-\frac{1}{2})$.

Takeaway: In a parabola, the squared expression identifies the axis direction, while $p$ determines the focus, directrix, and opening.

Ellipses: Axes, Vertices, and Foci

An ellipse is a closed curve with a center, vertices, co-vertices, major axis, minor axis, and foci. Its standard forms are

for a horizontal major axis, and

for a vertical major axis.

The center is $(h,k)$. The larger denominator is $a^2$, so it identifies the major-axis direction. The foci satisfy

For a horizontal ellipse, the vertices are $(h\pm a,k)$, the co-vertices are $(h,k\pm b)$, and the foci are $(h\pm c,k)$. For a vertical ellipse, the vertices are $(h,k\pm a)$, the co-vertices are $(h\pm b,k)$, and the foci are $(h,k\pm c)$. The major and minor axis lengths are $2a$ and $2b$, respectively.

For example, consider

The center is $(1,-2)$, with $a=5$ and $b=3$. Since the larger denominator is under the $x$-term, the major axis is horizontal. The vertices are $(-4,-2)$ and $(6,-2)$, while the co-vertices are $(1,-5)$ and $(1,1)$. Since $c^2=25-9=16$, $c=4$, so the foci are $(-3,-2)$ and $(5,-2)$.

Takeaway: Read the center from the translations, use the larger denominator to find the major-axis direction, and use $c^2=a^2-b^2$ to locate the foci.

Hyperbolas: Branches and Asymptotes

A hyperbola has two separate branches. Its standard horizontal form is

and its standard vertical form is

The positive squared term identifies the direction of opening. For a horizontal hyperbola, the vertices are $(h\pm a,k)$, the foci are $(h\pm c,k)$, and the asymptotes are

For a vertical hyperbola, the vertices are $(h,k\pm a)$, the foci are $(h,k\pm c)$, and the asymptotes are

The focal relationship is

For example, consider

The center is $(-3,1)$, and the positive $x$-term shows that the branches open left and right. Here, $a=4$ and $b=3$, so the vertices are $(-7,1)$ and $(1,1)$. The asymptotes are

Because $c^2=4^2+3^2=25$, $c=5$, and the foci are $(-8,1)$ and $(2,1)$.

To sketch a hyperbola, plot the center, construct a rectangle with half-widths $a$ and $b$, draw its diagonals as asymptotes, and sketch branches opening toward the vertices.

Takeaway: The positive term determines the opening direction, the vertices are $a$ units from the center, and the asymptotes guide the two branches.

Converting General Equations to Standard Form

Many conic equations become easier to interpret after rearranging terms and completing the square. The goal is to reach a standard equation whose translations and coefficients directly reveal the graph.

For a parabola, isolate the quadratic expression and complete the square in the squared variable. For example,

can be rearranged as

Add $9$ to both sides:

which gives

Therefore, the vertex is $(-2,3)$, the parabola opens right, and $p=1$.

For an ellipse, group the quadratic terms, complete the square in both variables, and divide so that the right-hand side is $1$. For example,

becomes

and therefore

This identifies an ellipse centered at $(1,-2)$ with a vertical major axis, $a=3$, and $b=2$.

For a general nonrotated equation

compare the signs and coefficients of the squared terms before doing detailed algebra. If the equation includes an $xy$ term,

the conic may be rotated, so ordinary completing-the-square steps may not provide the full orientation.

Takeaway: Classify first, complete the square second, and divide into standard form when possible. Standard form exposes the center or vertex, orientation, scale, foci, directrix, or asymptotes.

Comparing and Graphing Conics

The three principal noncircular conics can be compared by their shape, distance property, and main graphing features.

• Parabola

  • Branches: one.

  • Distance property: points are equally distant from a focus and a directrix.

  • Center: usually none.

  • Main graphing features: vertex and axis of symmetry.

  • Parameter relationship: uses $p$.

• Ellipse

  • Branches: one closed curve.

  • Distance property: the sum of distances to two foci is constant.

  • Center: yes.

  • Main graphing features: center, vertices, and co-vertices.

  • Parameter relationship: $c^2=a^2-b^2$.

• Hyperbola

  • Branches: two.

  • Distance property: the absolute difference of distances to two foci is constant.

  • Center: yes.

  • Main graphing features: center, vertices, and asymptotes.

  • Parameter relationship: $c^2=a^2+b^2$.

A reliable workflow is:

1. Inspect the squared terms and classify the conic.

2. Rearrange the equation and complete the square.

3. Convert to standard form.

4. Read the vertex or center and determine the orientation.

5. Calculate foci, directrix, co-vertices, or asymptotes as appropriate.

6. Plot the key points and sketch the curve symmetrically.

Final takeaway: Standard form is the bridge between an algebraic equation and a geometric graph. The signs, coefficients, translations, and parameter relationships together determine the conic's essential structure.

• asymptote