lesson on hyperbola

General Form and Identification
• The general equation for all conic sections: $Ax^2 + Cy^2 + Dx + Ey + F = 0$
• Identification by coefficients:
• Circles: $A = C$
• Ellipses: $A \neq C$ and AC > 0 (same sign)
• Hyperbolas: AC < 0 (opposite signs)
• Parabolas: Either $A = 0$ or $C = 0$, but not both
• For hyperbolas specifically, the critical property is that the product of coefficients $A$ times $C$ must be less than zero (negative)

Definition
• Conic sections are formed by the intersection of a plane and a cone.
• Each conic can be defined by the relationship of points to fixed features (foci, directrix, etc.).

Hyperbolas
Definition and Key Concept
• A hyperbola is the set of all points in a plane whose distance from two fixed points (called foci) has a constant difference.
• That difference equals $2a$.
• "I kind of think of a hyperbola as being an inside out ellipse. With a big boom."
• The defining property uses difference rather than sum (unlike ellipses): $|d<em>1 - d</em>2| = 2a$.

Standard Forms

1. Opens Left and Right (Horizontal)
   $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$
   • The positive term contains $(x-h)^2$.
   • Vertices are at $a$ units left and right from center.

2. Opens Up and Down (Vertical)
   $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$
   • The positive term contains $(y-k)^2$.
   • Vertices are at $a$ units above and below center.

Key Components and Terminology

• Center
  • Denoted as $(h, k)$ where $h$ and $k$ are the coordinates of the center point.
  • Located at the intersection of the transverse and conjugate axes.
  • Common mistake: Reading coordinates left to right instead of correctly identifying $x$ and $y$ values from the equation form.

• Vertices and Transverse Axis
  • Vertices: The two points where the hyperbola intersects the transverse axis.
  • Distance from center to each vertex: $a$ units.
  • Distance between both vertices: $2a$ units.
  • Transverse axis: The line segment connecting the two vertices.
  • Etymology: "Trans" means across (like transcontinental railroad); the transverse axis goes across from vertex to vertex.
  • This is analogous to the major axis of an ellipse.

• Foci
  • Two fixed points that define the hyperbola.
  • Distance from center to each focus: $c$ units.
  • Foci are located farther from the center than the vertices (always beyond the vertices).
  • For hyperbolas opening left/right: foci are on the horizontal transverse axis.
  • For hyperbolas opening up/down: foci are on the vertical transverse axis.

• Conjugate Axis
  • The line segment perpendicular to the transverse axis, passing through the center.
  • Distance from center to each conjugate axis endpoint: $b$ units.
  • Distance between both endpoints: $2b$ units.
  • The endpoints lie on the auxiliary rectangle used for graphing.

• Asymptotes
  • Lines that the hyperbola approaches but never touches.
  • The hyperbola gets increasingly close to the asymptotes as you move away from the center.
  • Help determine how "wide" or "narrow" the curves will be.
  • Can be found by drawing the auxiliary rectangle through vertices and conjugate endpoints, then drawing its diagonals.

Relationships Between $a$, $b$, and $c$
$a^2 + b^2 = c^2$
• This is the Pythagorean relationship for hyperbolas.
• $a$ is always the distance to vertices (underneath the positive term).
• $b$ is always the distance to conjugate axis endpoints (underneath the negative term).
• $c$ is always the distance to foci.
• Important: Unlike ellipses, the relationship uses addition, not subtraction.
• $c$ is always the largest of the three values: c > a and c > b.
• Between $a$ and $b$ themselves, either can be the larger number, or they can be equal.

Equation Structure and Signs
• "The $a$ squared always goes underneath the part that's not after the subtraction symbol, and the $b$ squared always comes after the subtraction symbol."
• The variable with the positive (non-subtracted) term determines the orientation.
• The variable coming after the subtraction symbol goes underneath $b^2$.
• Between $a$ and $b$ themselves, either could be the larger number or they could even be the same number.

Ellipses (for comparison)
Definition
• An ellipse is the set of all points in a plane whose distance from two fixed points (called foci) has a constant sum.
• That sum equals $2a$.
• Uses the property: $d<em>1 + d</em>2 = 2a$ (constant).

Standard Forms

1. Horizontal Orientation (Wider)
   $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$
   • Larger denominator is under (x-h)^2$.

2. Vertical Orientation (Taller)
   \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$<br>• Larger denominator is under$(y-k)^2$.

Key Difference from Hyperbola
• Uses addition throughout (no subtraction/negative term).
• $a$ represents the semi-major axis (always the larger semi-axis).
• $b$ represents the semi-minor axis (always the smaller semi-axis).
• Relationship: $a^2 = b^2 + c^2$ (note the arrangement).
• $a$ is always the largest of the three values.
• Foci are located inside the ellipse, between the center and vertices.

Circles and Parabolas
Circles
• A set of all points equally distant from a fixed center point.
• Distance equals the radius $r$.
• Standard form: $(x-h)^2 + (y-k)^2 = r^2$
• Center at $(h, k)$.
• Special case where $A = C$ in the general form.

Parabolas
• The set of all points equidistant from a fixed point (focus) and a fixed line (directrix).
• Equations vary based on direction of opening:
• Up/Down: $(x-h)^2 = 4p(y-k)$
• Left/Right: $(y-k)^2 = 4p(x-h)$
• $p$ determines the distance from vertex to focus and from vertex to directrix.

Converting General Form to Standard Form
Process Overview

1. Identify the conic type by examining the coefficients $A$ and $C$.

2. Rearrange terms grouping $x$ terms together and $y$ terms together.

3. Complete the square for both variable groups.

4. Factor out the leading coefficients so they equal 1 (critical step).

5. Divide through by the constant term to make the equation equal 1.

Completing the Square Details
• Key requirement: The coefficient of the squared term must be 1 before completing the square.
• To achieve this, factor out the coefficient from each variable group.
• "The process requires this to be a coefficient of 1. So to force it to be a 1, we'll factor out whatever number it actually is."
• For a group like $3(x^2 - 10x…)$:
• Half of the linear coefficient: $-10 \div 2 = -5$.
• Square it: $(-5)^2 = 25$.
• This value goes inside the parentheses and is multiplied by the factored coefficient outside.
• When factoring out a negative coefficient (like $-7$), carefully track the sign changes.
• Example: $-7(y^2 - 8y…)$ means inside we have $-8$, not $+8$.

Important Caution
• "Don't pre-determine that… because once we complete the square, that could have easily added up to a negative quantity… it might be true, and then the seeming contradiction might mess with your brain."
• Don't assume the orientation before completing the square; signs can flip when dividing.
• Always verify the final form matches what you expect for that conic type.

Verification Check
• In well-designed problems, the constant term often divides evenly into both denominators.
• If it doesn't divide evenly, this is a signal to triple-check your work.
• However, this is not absolute proof of error if the work is correct.

Graphing Hyperbolas
Step-by-Step Process

1. Identify Orientation
   • Determine which variable comes after the subtraction symbol.
   • If $y$ follows the minus: opens up and down (opens in the $y$-direction).
   • If $x$ follows the minus: opens left and right (opens in the $x$-direction).
   • This also determines whether to move vertically or horizontally from the center.

2. Plot the Center
   • Extract $(h, k)$ from the equation: $x - h$ and $y - k$.
   • Common error: confusing the sign; if the equation shows $(x+4)$, then $h = -4$.
   • Mark clearly on the graph and label it.

3. Plot Vertices
   • Vertices are $a$ units away from the center in the direction of opening.
   • Count (don't calculate with formulas): Count from the center $a$ units in the appropriate direction.
   • "Count or read off… Don't turn it into a formula. I guarantee you, if you're doing this problem on an ellipse, you click and ask for help. It's going to give you a formula. That formula doesn't work for every hyperbola. And it does not work for an ellipse. So counting works for everything."
   • Example: If center is $(-1, 4)$ and opening up/down with $a = 5$:
   • Vertices: $(-1, 4+5) = (-1, 9)$ and $(-1, 4-5) = (-1, -1)$.

4. Find Conjugate Axis Endpoints
   • Use the relationship $a^2 + b^2 = c^2$ to find $b$ if not directly given.
   • Plot $b$ units perpendicular to the transverse axis.
   • Example: If opening up/down, move $b$ units left and right from center.

5. Draw the Auxiliary Rectangle
   • Draw a rectangle using the vertices and conjugate axis endpoints as corners.
   • This rectangle helps establish the asymptote slopes.

6. Draw Asymptotes
   • Draw the diagonals of the auxiliary rectangle.
   • These diagonals pass through the center and establish the asymptotes.
   • The slopes of the asymptotes can be calculated from the rectangle:
   • For up/down opening: slope = $\pm \frac{a}{b}$.
   • For left/right opening: slope = $\pm \frac{b}{a}$.
   • Write equations using point-slope form: $y - k = m(x - h)$.

7. Sketch the Curves
   • The hyperbola must pass through the vertices.
   • The curves approach (but never touch) the asymptotes.
   • Critical: Marking vertices prevents accidental curve orientation errors.
   • The narrowness/width of the curves depends on how close the asymptotes are to each other.

Important Graphing Principles
• "Graph everything you have, because then when you look at it, you're probably going to make [correct conclusions about orientation]."
• Always mark and label key features: center, vertices, foci, asymptotes.
• Use a straight edge for asymptotes if possible to ensure accuracy.
• The asymptotes bound the region where the hyperbola curves appear.

Practical Examples and Problem-Solving Tips
Identifying Conic Type
• First examine $A \times C$ in the general form.
• Negative product → hyperbola.
• Same positive signs → ellipse.
• Equal coefficients → circle.
• One coefficient zero → parabola.
• "So that you can think about what the form is supposed to look like. So that when you get to your final answer, you're like oh yeah that's that's what the equation of a hyperbola looks like even if my numbers are wrong I still have a hyperbola equation."

Formula vs Counting Trade-offs
• Formulas are specific to orientations (one formula for up/down, different for left/right, doesn't apply to ellipses).
• Counting methods work universally for all shapes and orientations.
• Risk with formulas: Easy to accidentally use the wrong formula and get wrong orientation.

Common Mistakes to Avoid
• Confusing sign of $h$ and $k$ when extracting from $(x-h)$ or $(y-k)$ terms.
• Reading coordinates left-to-right rather than identifying $x$ and $y$ values correctly.
• Using ellipse terminology (major/minor axis) for hyperbolas.
• Pre-determining orientation before completing the square.
• Applying orientation-specific formulas universally.
• Forgetting that vertices come on the positive term, not the negative one.

Relationship Between $a$, $b$, and $c$
• All three are distances measured from the center.
• $a$ is always associated with the positive term (vertices).
• $b$ is always associated with the negative term (conjugate endpoints).
• $c$ is always the largest: $c^2 = a^2 + b^2$.
• Use Pythagorean relationship to find missing value