Conic Sections - Part 1: Standard Equations and Orientations

1️⃣ Review the four conic sections and their standard equations

2️⃣ Understand horizontal and vertical orientations

3️⃣ Learn about foci and their relationship to each conic

4️⃣ Distinguish between different equation forms

5️⃣ Apply knowledge to identify conic types quickly

1️⃣ Circle - The Symmetric Conic

The circle equation is the simplest of all conic sections:

Circle Equation

$$x^2 + y^2 = r^2$$

where r is the radius (not $r^2$!)

Key Points about Circles:

If the right side is 25, then $r = 5$ (not 25)
If the right side is 9, then $r = 3$ (not 9)
Both $x$ and $y$ have denominators of 1 (coefficients of 1)
All signs are positive
No horizontal or vertical orientation - a circle is always symmetric

2️⃣ Parabola - Single Variable Squared

The most important characteristic of a parabola is that only one variable is squared, and the other side contains $4c$.

Parabola Orientations

If $x^2$ is present: $y = 4cx$ → U-shape (opens up/down)

If $y^2$ is present: $x = 4cy$ → C-shape (opens left/right)

Direction Rules:

If we put a negative sign in front of $4c$, the parabola flips direction
U-shape opening up → becomes U-shape opening down
C-shape opening right → becomes C-shape opening left

3️⃣ Ellipse - Both Terms Positive

An ellipse has the standard equation where both terms on the left are positive and the right side equals 1.

Ellipse Equation

$$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$

The denominators are DIFFERENT!

Key Characteristics:

The denominators are different numbers
The larger number = $a^2$ (related to major axis)
The smaller number = $b^2$ (related to minor axis)
$a$ is associated with the major axis (longer)
$b$ is associated with the minor axis (shorter)

Ellipse Orientation Rules

If the larger number is under $x$:

→ Major axis is parallel to the $x$-axis

→ Ellipse is horizontally oriented

If the larger number is under $y$:

→ Major axis is parallel to the $y$-axis

→ Ellipse is vertically oriented

Special Case: If denominators are equal

→ No major axis, no minor axis

→ All radii become equal → We get a CIRCLE!

→ No $a^2$ or $b^2$ values → Circular shape

4️⃣ Hyperbola - The Contradiction

A hyperbola equation has one positive term and one negative term on the left side, with right side equal to 1. Notice the name suggests "excess" but the equation has a minus sign - this contradiction helps us remember!

Hyperbola Equation

$$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$

Notice the MINUS sign!

Important Rules:

$a^2$ goes under the POSITIVE term
$b^2$ goes under the NEGATIVE term
If $a$ and $b$ are equal, we do NOT get a circle (unlike ellipse)
The larger the difference between $a$ and $b$, the wider the hyperbola opens

Hyperbola Opening Direction

If the positive term is $\frac{x^2}{a^2}$:

→ Opens horizontally (C-shape and inverted C-shape)

→ Like ← → facing left and right

If the positive term is $\frac{y^2}{a^2}$:

→ Opens vertically (U-shape and inverted U-shape)

→ Like ↑ ↓ facing up and down

Quick Identification Guide

CIRCLE

$x^2 + y^2 = r^2$

Same coefficients, all positive

PARABOLA

Only ONE variable squared

Other side contains $4c$

ELLIPSE

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

Both positive, different denominators

HYPERBOLA

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

One positive, one negative

🎯 Key Takeaways from Part 1

Focus on standard equations first - they reveal the most information quickly
The right side is crucial: $r^2$ for circles, 1 for ellipses and hyperbolas
Look for orientation clues: which variable/term dominates
Remember the signs: all positive (circle/ellipse), mixed signs (hyperbola)
Understanding these basics prepares us for focus and directrix concepts in Part 2

All Conic Sections

All about Conic Sections Explained