Conic Sections

⭕ Introduction to Conic Sections

Part 1 of 7

What Are Conic Sections?

Conic sections are curves formed by intersecting a double cone with a plane:

General Second-Degree Equation

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

The discriminant $B^2 - 4AC$ determines the type:

• $B^2 - 4AC < 0$: ellipse (or circle if $A = C$, $B = 0$)
• $B^2 - 4AC = 0$: parabola
• $B^2 - 4AC > 0$: hyperbola

⭕ The Circle

Standard Form

$(x-h)^2 + (y-k)^2 = r^2$

Center $(h, k)$, radius $r$.

General Form

$x^2 + y^2 + Dx + Ey + F = 0$

Convert by completing the square.

Example: $x^2 + y^2 - 6x + 4y - 12 = 0$

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

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

$(x-3)^2 + (y+2)^2 = 25$

Center $(3, -2)$, radius $5$.

📐 Circle Properties

Key Facts

• All points are equidistant from the center
• A tangent line at point $P$ is perpendicular to the radius at $P$
• The equation of the tangent at $(x_0, y_0)$ on circle $x^2+y^2 = r^2$ is: $x_0 x + y_0 y = r^2$

Relationship to Other Conics

A circle is a special case of an ellipse where $a = b$ (both semi-axes equal).

Eccentricity of a circle: $e = 0$.

💡 Degenerate cases: If $r^2 < 0$ after completing the square, there is no real circle (empty set). If $r^2 = 0$, it is a single point.

Circle Quiz 🎯

Circle Computations 🧮

1) Center of $x^2 + y^2 - 10x + 6y + 9 = 0$: the $x$-coordinate of center = ?

2) Same circle: the $y$-coordinate of center = ?

3) Same circle: the radius = ?

Classify Conics 🔽

Exit Quiz ✅

🔵 The Ellipse

Part 2 of 7

Standard Form (Center at Origin)

Horizontal major axis: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b)$

Vertical major axis: $\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1 \quad (a > b)$

Key Elements

Defining Property

The sum of distances from any point on the ellipse to both foci is constant: $d_1 + d_2 = 2a$.

📝 Example: Analyze $\frac{x^2}{25} + \frac{y^2}{9} = 1$

$a^2 = 25, b^2 = 9 \implies a = 5, b = 3$

$c^2 = 25 - 9 = 16 \implies c = 4$

• Center: $(0, 0)$
• Vertices: $(\pm 5, 0)$
• Co-vertices: $(0, \pm 3)$
• Foci: $(\pm 4, 0)$
• Eccentricity: $e = \frac{4}{5} = 0.8$

Since $e$ is close to 1, this ellipse is fairly elongated.

Translated Ellipse

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

Same shape, centered at $(h, k)$ instead of the origin.

📊 Eccentricity: Shape of an Ellipse

Real-World Eccentricities

• Earth's orbit: $e \approx 0.017$ (nearly circular)
• Mars's orbit: $e \approx 0.093$
• Halley's comet: $e \approx 0.967$ (very elongated)
• Pluto: $e \approx 0.248$

💡 A higher eccentricity means the foci are farther from the center relative to the size of the ellipse.

Ellipse Quiz 🎯

Ellipse Calculations 🧮

For $\frac{x^2}{36}+\frac{y^2}{4}=1$:

1) $a$ = ?

2) $c$ = ? (Enter like "4sqrt2" if needed)

3) Eccentricity $e = \frac{c}{a}$ = ? (Enter as a fraction like "4/6" or simplify)

Ellipse Properties 🔽

Exit Quiz ✅

📐 The Parabola

Part 3 of 7

Definition

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

Standard Forms (Vertex at Origin)

💡 The value $p$ is the distance from the vertex to the focus (and also from the vertex to the directrix).

📝 Worked Examples

Example 1: Find focus and directrix of $x^2 = 12y$

Compare to $x^2 = 4py$: $4p = 12 \implies p = 3$

• Opens upward (positive coefficient)
• Focus: $(0, 3)$
• Directrix: $y = -3$
• Latus rectum (width through focus): $|4p| = 12$

Example 2: Write equation with focus at $(-2, 0)$

Focus on the negative $x$-axis → opens left → form $y^2 = -4px$

$p = 2$: $y^2 = -8x$

Translated Parabola

$(x-h)^2 = 4p(y-k)$

opens up/down, vertex at $(h, k)$.

$(y-k)^2 = 4p(x-h)$

opens right/left, vertex at $(h, k)$.

🔦 The Reflective Property

Parabolas have a remarkable property: any ray parallel to the axis reflects off the parabola and passes through the focus.

Applications

• Satellite dishes: Incoming parallel signals reflect to the focus (receiver)
• Car headlights: Light placed at the focus reflects outward in parallel beams
• Solar concentrators: Parallel sunlight focuses to a single point
• Suspension bridges: Cables under uniform load form parabolas

The focal length $p$ determines how "wide" or "narrow" the parabola opens. A small $p$ creates a narrow, tightly focused parabola.

Parabola Quiz 🎯

Parabola Calculations 🧮

1) For $x^2 = 16y$, find $p$: p = ?

2) For $y^2 = -24x$, the focus is at $(-a, 0)$. What is $a$?

3) A parabola opens upward with focus at $(0, 5)$. Its equation is $x^2 = ?y$. Enter the coefficient.

Parabola Properties 🔽

Exit Quiz ✅

📐 The Hyperbola

Part 4 of 7

Definition

A hyperbola is the set of all points where the difference of distances to two foci is constant: $|d_1 - d_2| = 2a$.

Standard Forms (Center at Origin)

Key relationship: $c^2 = a^2 + b^2$ (note the + sign, unlike the ellipse!)

📝 Worked Example

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

$a^2 = 9, b^2 = 16 \implies a = 3, b = 4$

$c^2 = 9 + 16 = 25 \implies c = 5$

• Center: $(0, 0)$
• Vertices: $(\pm 3, 0)$
• Foci: $(\pm 5, 0)$
• Asymptotes: $y = \pm\frac{4}{3}x$
• Eccentricity: $e = \frac{c}{a} = \frac{5}{3} \approx 1.67$

Graphing Strategy

1. Plot vertices at $(\pm a, 0)$
2. Mark co-vertices at $(0, \pm b)$
3. Draw the central rectangle through these four points
4. Draw asymptotes as diagonals of this rectangle
5. Sketch the hyperbola approaching but never touching the asymptotes

🧠 Hyperbola vs. Ellipse

Eccentricity of Hyperbolas

• $e$ close to $1$: branches open very narrowly
• $e = \sqrt{2}$: rectangular hyperbola ($a = b$)
• $e$ large: branches open very widely

💡 The conjugate axis has length $2b$; the transverse axis has length $2a$ and connects the vertices.

Hyperbola Quiz 🎯

Hyperbola Calculations 🧮

For $\frac{y^2}{36}-\frac{x^2}{64}=1$:

1) $a$ = ?

2) $c$ = ?

3) Eccentricity $e$ = ? (Enter as a fraction)

Hyperbola Properties 🔽

Exit Quiz ✅

🔄 Rotated Conics & General Second-Degree Equations

Part 5 of 7

The General Second-Degree Equation

$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$

When $B \neq 0$, the conic is rotated — its axes are not aligned with the coordinate axes.

The Discriminant Test (Revisited)

$\Delta = B^2 - 4AC$ determines the type:

💡 The discriminant is invariant under rotation — changing coordinates doesn't change $B^2-4AC$.

🔀 Eliminating the $xy$-Term

To remove the $Bxy$ term, rotate axes by angle $\theta$ where:

$\cot 2\theta = \frac{A-C}{B}$

Rotation Formulas

$x = X\cos\theta - Y\sin\theta$ $y = X\sin\theta + Y\cos\theta$

Substituting transforms $Ax^2+Bxy+Cy^2+\ldots = 0$ into:

$A'X^2 + C'Y^2 + D'X + E'Y + F' = 0$

with no $XY$-term — now in standard position!

Example

$xy = 1$ has $A=0, B=1, C=0$. $\cot 2\theta = 0 \implies \theta = 45°$.

After rotation: $\frac{X^2}{2} - \frac{Y^2}{2} = 1$ — a hyperbola!

🏷️ Classification Practice

Example 1: $x^2 + 4xy + 4y^2 + 2x - 3y + 1 = 0$

$A=1, B=4, C=4$

$\Delta = 16 - 16 = 0$ → Parabola

Example 2: $2x^2 + 3xy - 2y^2 + x - 5 = 0$

$A=2, B=3, C=-2$

$\Delta = 9 - 4(2)(-2) = 9+16 = 25 > 0$ → Hyperbola

Example 3: $3x^2 + 2xy + 3y^2 - 8 = 0$

$A=3, B=2, C=3$

$\Delta = 4 - 36 = -32 < 0$ → Ellipse

Note: Degenerate cases (empty set, single point, intersecting lines) can occur when the equation factors.

Classification Quiz 🎯

Discriminant Practice 🧮

Calculate $\Delta = B^2-4AC$ for each:

1) $5x^2+3xy+2y^2=10$: $\Delta$ = ?

2) $x^2-4xy+4y^2+y=0$: $\Delta$ = ?

3) $2x^2+5xy-3y^2=1$: $\Delta$ = ?

Rotated Conics Concepts 🔽

Exit Quiz ✅

🌍 Applications of Conic Sections

Part 6 of 7

Conics appear throughout science, engineering, and nature:

Planetary Orbits (Ellipses)

Kepler's First Law: planets orbit the Sun in ellipses with the Sun at one focus.

Satellite Dishes & Headlights (Parabolas)

The reflective property of parabolas focuses signals to a single point or projects light in parallel beams.

Hyperbolic Navigation (Hyperbolas)

LORAN (Long Range Navigation) uses the difference in signal arrival times from two stations, which traces a hyperbola.

Architecture (All Conics)

• Elliptical rooms (whispering galleries)
• Parabolic arches and bridges
• Hyperbolic cooling towers

🪐 Orbital Mechanics

A body under gravity follows a conic section. The type depends on its energy:

Example: Earth's Orbit

$a = 1.496 \times 10^8$ km, $e = 0.0167$

• Perihelion (closest): $a(1-e) = 1.471 \times 10^8$ km
• Aphelion (farthest): $a(1+e) = 1.521 \times 10^8$ km

The nearly circular orbit ($e \approx 0$) gives us relatively stable seasons.

🔊 Acoustic & Optical Applications

Whispering Gallery (Ellipse)

In an elliptical room, sound from one focus reflects off the wall and converges at the other focus. Famous examples:

• St. Paul's Cathedral, London
• National Statuary Hall, U.S. Capitol

Parabolic Reflectors

A parabolic mirror reflects all incoming parallel rays to the focus:

• Telescopes (reflecting telescopes)
• Solar cookers
• Microphone dishes (for recording distant sounds)

Hyperbolic Mirrors

Used in Cassegrain telescopes: the secondary mirror is hyperbolic, redirecting light from the primary parabolic mirror to a more convenient focal point.

Applications Quiz 🎯

Application Calculations 🧮

1) A planet orbits with $a = 10$ AU and $e = 0.6$. Perihelion $= a(1-e) =$ ? AU

2) Same planet: aphelion $= a(1+e) =$ ? AU

3) A parabolic dish has equation $x^2 = 8y$ (in feet). The receiver should be placed at the focus: $y =$ ? feet

Real-World Conics 🔽

Exit Quiz ✅

🎯 Conic Sections — Full Synthesis

Part 7 of 7

Summary of All Conics

The $c$-Relationship

• Ellipse: $c^2 = a^2 - b^2$ (foci inside)
• Hyperbola: $c^2 = a^2 + b^2$ (foci outside)
• Parabola: just one focus at distance $p$ from vertex

🗺️ Identification Strategy

Given a second-degree equation, follow this flowchart:

Step 1: Is there an $xy$-term?

• Yes → Use discriminant $B^2-4AC$ to classify; rotate if needed
• No → Go to Step 2

Step 2: Which squared terms are present?

• Both $x^2$ and $y^2$ → Go to Step 3
• Only $x^2$ or only $y^2$ → Parabola

Step 3: Are the coefficients of $x^2$ and $y^2$ the same sign?

• Same sign, same value → Circle
• Same sign, different values → Ellipse
• Opposite signs → Hyperbola

Converting to Standard Form

1. Group $x$-terms and $y$-terms
2. Complete the square for each variable
3. Divide to get $1$ on the right side

📝 Comprehensive Example

Identify and graph: $4x^2+9y^2-16x+54y+61=0$

Step 1: Group and complete the square.

$4(x^2-4x) + 9(y^2+6y) = -61$

$4(x^2-4x+4) + 9(y^2+6y+9) = -61+16+81$

$4(x-2)^2 + 9(y+3)^2 = 36$

Step 2: Divide by 36.

$\frac{(x-2)^2}{9} + \frac{(y+3)^2}{4} = 1$

Identify: Ellipse, center $(2,-3)$, $a=3$, $b=2$, horizontal major axis.

$c = \sqrt{9-4} = \sqrt{5}$, foci at $(2\pm\sqrt{5}, -3)$.

Synthesis Quiz 🎯

Complete the Square 🧮

Convert $x^2+4y^2+2x-24y+33=0$ to standard form.

1) Center $h$ = ?

2) Center $k$ = ?

3) This is a(n): (type "ellipse", "parabola", "hyperbola", or "circle")

Master Classification 🔽

Final Exit Quiz ✅