Conic Sections: Circles Ellipses Parabolas Hyperbolas

Take a double cone—two cones stacked tip-to-tip. Now slice through it with a plane.

Different angles give different curves.

Horizontal slice: circle. Tilted slice: ellipse. Slice parallel to the cone's edge: parabola. Steep slice through both cones: hyperbola.

These are the conic sections.

Four curves, one geometric origin. And each has an algebraic equation.

The Unlock: Geometry Becomes Algebra

The ancient Greeks studied conic sections purely geometrically. They sliced cones and studied the resulting curves.

Then Descartes connected geometry to algebra. Every curve could be described by an equation.

Conic sections bridge the two worlds.

Each conic is a geometric shape and an algebraic equation. The geometry gives intuition. The algebra gives precision.

The Four Conics

Circle: All points equidistant from a center.

Ellipse: All points such that the sum of distances to two foci is constant.

Parabola: All points equidistant from a focus and a directrix.

Hyperbola: All points such that the difference of distances to two foci is constant.

These definitions come from geometry. But each translates to an equation.

The Circle: x² + y² = r²

A circle centered at the origin with radius r:

x² + y² = r².

Every point (x, y) on the circle satisfies this equation.

Example: x² + y² = 25.

This is a circle centered at (0, 0) with radius 5.

Shifted circle, center (h, k):

(x - h)² + (y - k)² = r².

Example: (x - 3)² + (y + 2)² = 16.

Center: (3, -2). Radius: 4.

The Ellipse: x²/a² + y²/b² = 1

An ellipse centered at the origin:

x²/a² + y²/b² = 1.

The ellipse has two axes: the major axis (longer) and the minor axis (shorter).

If a > b: the major axis is horizontal, length 2a. The minor axis is vertical, length 2b.

If b > a: the major axis is vertical, length 2b. The minor axis is horizontal, length 2a.

Example: x²/9 + y²/4 = 1.

a² = 9, so a = 3. b² = 4, so b = 2.

Horizontal major axis, length 6. Vertical minor axis, length 4.

Foci of the Ellipse

An ellipse has two foci (singular: focus).

For an ellipse centered at the origin with horizontal major axis:

x²/a² + y²/b² = 1 (a > b).

The foci are at (±c, 0), where c² = a² - b².

Example: x²/25 + y²/9 = 1.

a² = 25, b² = 9. c² = 25 - 9 = 16, so c = 4.

Foci: (4, 0) and (-4, 0).

The sum of distances from any point on the ellipse to the two foci is constant (equal to 2a).

The Parabola: y = ax²

The simplest parabola opens upward or downward:

y = ax².

If a > 0, it opens upward. If a < 0, it opens downward.

Vertex at the origin. Axis of symmetry along the y-axis.

Shifted parabola:

y = a(x - h)² + k.

Vertex at (h, k).

Example: y = 2(x - 1)² + 3.

Vertex: (1, 3). Opens upward (a = 2 > 0).

Horizontal parabola (opens left or right):

x = a(y - k)² + h.

Focus and Directrix of the Parabola

A parabola is defined as the set of points equidistant from a focus and a directrix (a line).

For y = ax²:

Focus: (0, 1/(4a)).

Directrix: y = -1/(4a).

Example: y = x².

a = 1. Focus: (0, 1/4). Directrix: y = -1/4.

Every point on the parabola is the same distance from (0, 1/4) and the line y = -1/4.

The Hyperbola: x²/a² - y²/b² = 1

A hyperbola centered at the origin with horizontal transverse axis:

x²/a² - y²/b² = 1.

The hyperbola has two branches, opening left and right.

The vertices are at (±a, 0).

For a hyperbola with vertical transverse axis:

y²/a² - x²/b² = 1.

The branches open up and down. Vertices at (0, ±a).

Foci of the Hyperbola

For x²/a² - y²/b² = 1:

The foci are at (±c, 0), where c² = a² + b².

Example: x²/9 - y²/16 = 1.

a² = 9, b² = 16. c² = 9 + 16 = 25, so c = 5.

Foci: (5, 0) and (-5, 0).

The difference of distances from any point on the hyperbola to the two foci is constant (equal to 2a).

Asymptotes of the Hyperbola

A hyperbola has two asymptotes—lines the branches approach but never touch.

For x²/a² - y²/b² = 1:

Asymptotes: y = ±(b/a)x.

Example: x²/9 - y²/16 = 1.

a = 3, b = 4. Asymptotes: y = ±(4/3)x.

The branches of the hyperbola get closer and closer to these lines as |x| and |y| increase.

General Conic Equation

All conic sections fit into a single general equation:

Ax² + Bxy + Cy² + Dx + Ey + F = 0.

The type of conic depends on A, B, and C.

Discriminant: B² - 4AC.

• B² - 4AC < 0: Ellipse (or circle if A = C and B = 0).
• B² - 4AC = 0: Parabola.
• B² - 4AC > 0: Hyperbola.

This classifies conics algebraically.

Completing the Square

To identify and graph a conic, you often need to complete the square.

Example: x² + y² - 4x + 6y - 3 = 0.

Group x terms and y terms:

(x² - 4x) + (y² + 6y) = 3.

Complete the square:

(x² - 4x + 4) + (y² + 6y + 9) = 3 + 4 + 9.

(x - 2)² + (y + 3)² = 16.

This is a circle centered at (2, -3) with radius 4.

Eccentricity

Eccentricity (e) measures how "stretched" a conic is.

• Circle: e = 0.
• Ellipse: 0 < e < 1.
• Parabola: e = 1.
• Hyperbola: e > 1.

For an ellipse: e = c/a, where c is the distance from center to focus.

For a hyperbola: e = c/a.

Eccentricity is a single number that characterizes the conic's shape.

Conic Sections in Orbits

Planetary orbits are ellipses, with the sun at one focus. (Kepler's first law.)

Comets with enough energy escape the solar system on hyperbolic trajectories.

Objects dropped near Earth follow parabolic paths (ignoring air resistance).

Conic sections describe motion under an inverse-square force (gravity).

Conic Sections in Optics

Parabolic mirrors focus parallel light rays to a single point (the focus). This is why satellite dishes and telescopes use parabolic shapes.

Elliptical mirrors reflect light from one focus to the other. Whispering galleries use this property.

Hyperbolic mirrors are used in telescope designs (Cassegrain telescopes).

Conics are central to lens and mirror design.

Conic Sections in Engineering

Parabolic arches distribute weight efficiently. They appear in bridges and architecture.

Elliptical gears create non-uniform rotational motion, used in machinery.

Hyperbolic cooling towers use the strength of hyperbolic shapes.

Conics aren't just mathematical curiosities. They're structural and functional.

Parametric Equations for Conics

Conics can be expressed parametrically.

Circle: x = r cos(t) y = r sin(t)

Ellipse: x = a cos(t) y = b sin(t)

Parabola: x = t y = t²

Hyperbola: x = a sec(t) y = b tan(t)

Parametric forms make it easy to trace the curve as t varies.

Polar Equations for Conics

In polar coordinates (r, θ), a conic with one focus at the origin:

r = ed / (1 + e cos(θ)).

• e = eccentricity.
• d = distance from focus to directrix.

For e < 1: ellipse. For e = 1: parabola. For e > 1: hyperbola.

Polar form is natural for orbital mechanics.

Degenerate Conics

Some slices give degenerate cases:

• Slice through the cone's tip: a point.
• Slice along the edge of the cone: a line.
• Slice through both cones at the tip: two intersecting lines.

These are technically conics, but they're edge cases.

Rotating Conics: The Bxy Term

If the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 has B ≠ 0, the conic is rotated.

The xy term represents a rotation of the axes.

To eliminate it, rotate the coordinate system by an angle θ:

tan(2θ) = B / (A - C).

After rotation, the equation becomes Ax'² + Cy'² + ... = 0 (no x'y' term).

This is how you identify and graph rotated conics.

Why Four Conics?

Why exactly four curves from slicing a cone?

It comes down to the angle of the slice relative to the cone's slope.

• Perpendicular to the axis: circle.
• Tilted, but not parallel to the edge: ellipse.
• Parallel to the edge: parabola.
• Steeper than the edge, cutting both cones: hyperbola.

Each corresponds to a different relationship between the plane and the cone's geometry.

Conics as Quadratic Curves

Algebraically, conics are quadratic curves: the highest degree is 2.

Ax² + Bxy + Cy² + Dx + Ey + F = 0.

This makes them solvable. You can find intersections, tangents, areas, arc lengths (with some effort).

Higher-degree curves (cubics, quartics) are messier. Conics are the last "nice" curves before things get complicated.

Tangent Lines to Conics

To find the tangent line to a conic at a point, use implicit differentiation.

Example: x²/9 + y²/4 = 1.

Differentiate implicitly:

(2x)/9 + (2y/4)(dy/dx) = 0.

Solve for dy/dx:

dy/dx = -4x / (9y).

At the point (3, 0): slope is undefined (vertical tangent).

At the point (0, 2): slope is 0 (horizontal tangent).

Conic Sections and Projective Geometry

In projective geometry, all conics are equivalent. A circle, ellipse, parabola, and hyperbola are all the "same" curve, just viewed from different perspectives.

This unifies the four conics into a single object.

Projective geometry is abstract, but it reveals deep structure.

Why Conics Matter in Precalculus

Conics connect algebra and geometry.

They're examples of implicit equations (not y = f(x), but a relationship between x and y).

They require completing the square—a key algebraic skill.

They introduce foci, directrices, eccentricity—geometric concepts with algebraic expressions.

And they're the first curves beyond polynomials that students encounter systematically.

Common Mistakes

Mistake 1: Confusing a and b in ellipses.

a is the semi-major axis (longer). b is the semi-minor axis (shorter). But which is horizontal depends on the equation.

Mistake 2: Forgetting the difference in the hyperbola formula.

Ellipse: x²/a² + y²/b² = 1 (plus sign).

Hyperbola: x²/a² - y²/b² = 1 (minus sign).

The sign determines the conic type.

Mistake 3: Misidentifying the center.

For (x - h)² + (y - k)² = r², the center is (h, k), not (-h, -k).

Mistake 4: Confusing foci and vertices.

Vertices are the closest points on the conic to the center. Foci are inside (for ellipse) or outside (for hyperbola).

Mistake 5: Ignoring asymptotes for hyperbolas.

Asymptotes guide the shape. Sketch them first, then draw the branches.

The Payoff: Seeing Curves as Transformations of Circles

Circles are simple. Conics are circles transformed.

An ellipse is a circle stretched in one direction.

A parabola is the limit as one focus moves to infinity.

A hyperbola is a circle reflected and split.

When you see conics this way, they're not four disconnected curves. They're variations on a theme.

That's the conceptual shift: from memorizing formulas to seeing geometric relationships.

Part 7 of the Precalculus series.

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