AP Precalculus: Conic Sections

Master parabolas, circles, ellipses, and hyperbolas with formulas and properties

🥎 Parabola ⭕ Circle 🥚 Ellipse 🔗 Hyperbola

📚 Understanding Conic Sections

Conic sections are curves formed by the intersection of a plane and a double cone. The four types—parabola, circle, ellipse, and hyperbola—each have unique properties and equations. They appear throughout physics, astronomy, engineering, and architecture.

1 Parabola

A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Eccentricity \(e = 1\).

Vertical Parabola (opens up/down)

\(y = a(x - h)^2 + k\)

Opens up if \(a > 0\)
Opens down if \(a < 0\)

Horizontal Parabola (opens left/right)

\(x = a(y - k)^2 + h\)

Opens right if \(a > 0\)
Opens left if \(a < 0\)

Key Properties (Vertical Parabola)

Vertex

\((h, k)\)

Focus

\(\left(h, k + \frac{1}{4a}\right)\)

Directrix

\(y = k - \frac{1}{4a}\)

Axis of Symmetry

\(x = h\)

Focal Length

\(p = \frac{1}{4a}\)

Latus Rectum

\(|4p| = \frac{1}{|a|}\)

📌 Example

Given: \(y = 2(x - 3)^2 + 1\)

Vertex: \((3, 1)\)

Focus: \(p = \frac{1}{4(2)} = \frac{1}{8}\), so focus at \(\left(3, 1 + \frac{1}{8}\right) = \left(3, \frac{9}{8}\right)\)

Directrix: \(y = 1 - \frac{1}{8} = \frac{7}{8}\)

2 Circle

A circle is the set of all points equidistant from a fixed point (center). Eccentricity \(e = 0\).

Standard Form \((x - h)^2 + (y - k)^2 = r^2\)

Center

\((h, k)\)

Radius

\(r\)

Diameter

\(2r\)

Eccentricity

\(e = 0\)

📌 Example

Given: \((x + 2)^2 + (y - 5)^2 = 16\)

Center: \((-2, 5)\) — note the signs!

Radius: \(r = \sqrt{16} = 4\)

💡 General Form

\(x^2 + y^2 + Dx + Ey + F = 0\) can be converted to standard form by completing the square.

3 Ellipse

An ellipse is the set of all points where the sum of distances to two foci is constant. Eccentricity \(0 < e < 1\).

Horizontal Major Axis

\(\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1\)

where \(a > b\), major axis along x

Vertical Major Axis

\(\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1\)

where \(a > b\), major axis along y

Key Properties

Center

\((h, k)\)

Vertices

\((h \pm a, k)\) or \((h, k \pm a)\)

Co-vertices

\((h, k \pm b)\) or \((h \pm b, k)\)

Foci

\((h \pm c, k)\) or \((h, k \pm c)\)

Major Axis Length

\(2a\)

Minor Axis Length

\(2b\)

Relationship Between a, b, and c \(c^2 = a^2 - b^2\) (for ellipse, \(c < a\))

Eccentricity \(e = \frac{c}{a}\), where \(0 < e < 1\)

📌 Example

Given: \(\frac{(x-1)^2}{25} + \frac{(y+2)^2}{9} = 1\)

Center: \((1, -2)\)

Since \(25 > 9\): \(a^2 = 25\), \(b^2 = 9\), so \(a = 5\), \(b = 3\)

Major axis: Horizontal (along x)

Vertices: \((1 \pm 5, -2) = (-4, -2)\) and \((6, -2)\)

Find c: \(c = \sqrt{25 - 9} = \sqrt{16} = 4\)

Foci: \((1 \pm 4, -2) = (-3, -2)\) and \((5, -2)\)

4 Hyperbola

A hyperbola is the set of all points where the difference of distances to two foci is constant. Eccentricity \(e > 1\).

Horizontal Transverse Axis

\(\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1\)

Opens left and right

Vertical Transverse Axis

\(\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1\)

Opens up and down

Key Properties

Center

\((h, k)\)

Vertices

\((h \pm a, k)\) or \((h, k \pm a)\)

Foci

\((h \pm c, k)\) or \((h, k \pm c)\)

Transverse Axis

Length = \(2a\)

Conjugate Axis

Length = \(2b\)

Eccentricity

\(e = \frac{c}{a} > 1\)

Relationship Between a, b, and c \(c^2 = a^2 + b^2\) (for hyperbola, \(c > a\))

Asymptotes Horizontal: \(y - k = \pm\frac{b}{a}(x - h)\) Vertical: \(y - k = \pm\frac{a}{b}(x - h)\)

📌 Example

Given: \(\frac{(x+1)^2}{16} - \frac{(y-3)^2}{9} = 1\)

Center: \((-1, 3)\)

Since x² term is positive: Horizontal transverse axis

Values: \(a^2 = 16\), \(b^2 = 9\), so \(a = 4\), \(b = 3\)

Vertices: \((-1 \pm 4, 3) = (-5, 3)\) and \((3, 3)\)

Find c: \(c = \sqrt{16 + 9} = \sqrt{25} = 5\)

Foci: \((-1 \pm 5, 3) = (-6, 3)\) and \((4, 3)\)

Asymptotes: \(y - 3 = \pm\frac{3}{4}(x + 1)\)

⚠️ Ellipse vs. Hyperbola

Ellipse: \(c^2 = a^2 - b^2\) (subtraction). Hyperbola: \(c^2 = a^2 + b^2\) (addition). Don't mix them up!

5 Converting to Standard Form

General form: \(Ax^2 + By^2 + Cx + Dy + E = 0\). Convert by completing the square.

Steps to Convert

Group x terms and y terms together
Factor out coefficients of \(x^2\) and \(y^2\) if not 1
Complete the square for each variable group
Add the same values to both sides (accounting for factored coefficients)
Write in standard form and divide if needed

📌 Example: Circle

Convert: \(x^2 + y^2 - 6x + 4y - 12 = 0\)

Group: \((x^2 - 6x) + (y^2 + 4y) = 12\)

Complete squares: \((x^2 - 6x + 9) + (y^2 + 4y + 4) = 12 + 9 + 4\)

Standard form: \((x - 3)^2 + (y + 2)^2 = 25\)

Result: Circle with center \((3, -2)\) and radius 5

6 Identifying Conic Type

Given general form \(Ax^2 + By^2 + Cx + Dy + E = 0\), identify the conic by comparing coefficients A and B.

Condition	Conic Type
\(A = B\) (and both ≠ 0)	Circle
\(A \neq B\) but same sign	Ellipse
\(A\) and \(B\) have opposite signs	Hyperbola
Only one of \(A\) or \(B\) is 0	Parabola

💡 Quick Check

Look at the signs and coefficients of \(x^2\) and \(y^2\). Same coefficient = circle, same sign different coefficient = ellipse, opposite signs = hyperbola, missing one = parabola.

📋 Quick Reference

Conic	Eccentricity	c² Formula	Key Feature
Circle	\(e = 0\)	—	All points equidistant from center
Ellipse	\(0 < e < 1\)	\(c^2 = a^2 - b^2\)	Sum of distances to foci = constant
Parabola	\(e = 1\)	—	Equidistant from focus & directrix
Hyperbola	\(e > 1\)	\(c^2 = a^2 + b^2\)	Difference of distances to foci = constant

📚 Understanding Conic Sections

1 Parabola

Key Properties (Vertical Parabola)

2 Circle

3 Ellipse

Key Properties

4 Hyperbola

Key Properties

5 Converting to Standard Form

Steps to Convert

6 Identifying Conic Type

📋 Quick Reference

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