Conic Sections - AP PreCalculus Study Guide

The Core Idea: Conic Sections

A single, general algebraic equation can describe four distinct and important geometric shapes: the circle, parabola, ellipse, and hyperbola. The equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ is the general form that encompasses all of these shapes, which are collectively known as conic sections.

The central task of this topic is to bridge the gap between the algebra of this equation and the geometry of the resulting shape. This involves two key skills: first, learning to identify which of the four conic sections an equation represents by analyzing its coefficients; and second, extracting the specific, defining geometric features—such as the center, focus, or vertices—that give each conic its unique size, position, and orientation in the coordinate plane.

Key Rules for Identification

The type of conic section can be determined by examining the coefficients of the squared terms ($x^2$ and $y^2$) in the general equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$. For the purposes of AP Precalculus, we primarily analyze equations where $B=0$.

• Circle: An equation represents a circle if the coefficients of $x^2$ and $y^2$ are equal and non-zero.

  • Condition: $A=C$ and $A,C\ne 0$.

  • Example: $3x^2+3y^2-6x+12y-2=0$

• Parabola: An equation represents a parabola if exactly one of the variables is squared.

  • Condition: Either $A=0$ or $C=0$, but not both.

  • Example: $x^2-4x+8y-20=0$ (contains $x^2$ but not $y^2$)

• Ellipse: An equation represents an ellipse if the coefficients of $x^2$ and $y^2$ have the same sign but are not equal.

  • Condition: $A\cdot C>0$ and $A\ne C$.

  • Example: $9x^2+4y^2+18x-16y-11=0$

• Hyperbola: An equation represents a hyperbola if the coefficients of $x^2$ and $y^2$ have opposite signs.

  • Condition: $A\cdot C<0$.

  • Example: $9x^2-16y^2-36x-32y-124=0$

Understanding the Equation-Shape Connection

The relationship between the $A$ and $C$ coefficients in the general equation directly dictates the fundamental geometry of the conic section.

• A parabola has only one squared term, which results in a curve that is unbounded in one direction, creating its familiar "U" shape. The linear term controls its orientation and width.

• A circle requires $A=C$ because this equality ensures the shape is perfectly symmetric in all directions from its center. Any change in $x$ has an identical effect as a change in $y$, producing a constant radius.

• An ellipse has $A$ and $C$ with the same sign but different values. This inequality causes the shape to be stretched or compressed along one axis relative to the other, forming a major (longer) axis and a minor (shorter) axis.

• A hyperbola has $A$ and $C$ with opposite signs. This fundamental opposition in the equation causes the graph to split into two separate, opposing branches that extend infinitely, guided by asymptotes.

Core Concepts & Rules

• The equation $A{x}^2+Bxy+C{y}^2+Dx+Ey+F=0$ is the general algebraic representation of any conic section.

• The key characteristics that define a circle are its center point and the length of its radius.

• The key characteristics that define a parabola are its vertex (the turning point), its focus (a special point that defines the curve), and its directrix (a special line that defines the curve).

• The key characteristics that define an ellipse are its center, its two foci, and the lengths of its major and minor axes.

• The key characteristics that define a hyperbola are its center, its two foci, its two vertices, and its asymptotes (the lines the curve approaches).

Step-by-Step Example 1: Identifying a Conic Section

Problem: Identify the type of conic section represented by the equation $5x^2+20y^2-10x+80y+65=0$.

Step 1: Analyze the squared terms

Examine the terms containing $x^2$ and $y^2$ in the given equation.

The terms are $5x^2$ and $20y^2$.

Step 2: Identify the coefficients A and C

From the general form $A{x}^2+C{y}^2+Dx+Ey+F=0$, we can identify the coefficients of the squared terms.

• $A=5$ (the coefficient of $x^2$)

• $C=20$ (the coefficient of $y^2$)

Step 3: Compare the coefficients

Apply the rules for identification based on $A$ and $C$.

• Do $A$ and $C$ have the same sign? Yes, both are positive.

• Are $A$ and $C$ equal? No, $5\ne 20$.

Step 4: State the conclusion

Because the coefficients of the squared terms have the same sign but are not equal, the equation represents an ellipse.

Step-by-Step Example 2: Identifying Key Characteristics of a Hyperbola

Problem: A hyperbola is given by the equation $$\frac{(y+2{)}^2}{9}-\frac{(x-5{)}^2}{16}=1$$. Identify its center, vertices, foci, and asymptotes.

Step 1: Identify the center (h, k)

The center is given by the values subtracted from $x$ and $y$.

• The center is at $(h,k)=(5,-2)$.

Step 2: Determine the orientation and vertices

The positive term determines the orientation of the transverse axis (the axis that connects the vertices). Here, the $y^2$ term is positive, so the hyperbola opens vertically (up and down).

• $a^2=9$, so $a=3$. The vertices are $a$ units above and below the center.

• Vertices: $(5,-2\pm 3)$, which are $(5,1)$ and $(5,-5)$.

Step 3: Find the foci

For a hyperbola, the distance from the center to each focus is $c$, where $c^2=a^2+b^2$.

• $b^2=16$, so $b=4$.

• $c^2=9+16=25$, which means $c=5$.

• The foci are $c$ units above and below the center (along the same axis as the vertices).

• Foci: $(5,-2\pm 5)$, which are $(5,3)$ and $(5,-7)$.

Step 4: Determine the equations of the asymptotes

The asymptotes are lines that pass through the center `(h, k)$withslopesof$\pm \frac{a}{b}$foraverticalhyperbola.-Slope:$\pm \frac{a}{b} = \pm \frac{3}{4}. - Using point-slope form $y - y_1 = m(x - x_1):

-   $y-(-2)=\frac{3}{4}(x-5)⟹y+2=\frac{3}{4}(x-5)$

-   $y-(-2)=-\frac{3}{4}(x-5)⟹y+2=-\frac{3}{4}(x-5)$

Step 5: Summarize the key characteristics

• Center:$(5,-2)$

• Vertices:$(5,1)$ and $(5,-5)$

• Foci:$(5,3)$ and $(5,-7)$

• Asymptotes:$y+2=\pm \frac{3}{4}(x-5)$

Using Your Calculator

A graphing calculator cannot directly graph a conic from its general form because it is not a function. To visualize a conic, you must first solve its equation for $y$. This process typically results in two separate functions that represent the top and bottom (or left and right) halves of the curve.

Example: Graphing the ellipse $\frac{(x-1{)}^2}{16}+\frac{y^2}{4}=1$

Step 1: Isolate the $y^2$ term

$$\frac{y^2}{4}=1-\frac{(x-1{)}^2}{16}$$

$$y^2=4\left(1-\frac{(x-1{)}^2}{16}\right)$$

$$y^2=4-\frac{4(x-1{)}^2}{16}=4-\frac{(x-1{)}^2}{4}$$

Step 2: Take the square root of both sides

Remember to include both the positive and negative roots.

$$y=\pm \sqrt{4-\frac{(x-1{)}^2}{4}}$$

Step 3: Enter the two equations into the calculator

In the Y= editor, enter the two halves of the ellipse:

• $Y_1=\sqrt{4-(x-1{)}^2/4}$

• $Y_2=-\sqrt{4-(x-1{)}^2/4}$

Step 4: Adjust the viewing window

Pressing GRAPH may produce a distorted image. To ensure the aspect ratio is correct and circles look like circles, use the zoom square feature. On a TI-84, this is ZOOM -> 5:ZSquare. This will provide an accurate visual representation of the ellipse, which you can use to check the location of its center and the lengths of its axes.

AP Exam Quick Hit

Common Question Types

• Identification from General Form: You will be given an equation in the form $A{x}^2+C{y}^2+Dx+Ey+F=0$ and asked to identify the type of conic section it represents.

  • Example: "Which of the following conic sections is represented by the equation $2x^2-5x+3y-2=0$?" (Answer: Parabola, because only $x$ is squared).

• Finding Characteristics from Standard Form: You will be given an equation in a standard, recognizable form and asked to identify a specific key feature.

  • Example: "What are the coordinates of the center of the ellipse given by $\frac{(x+1{)}^2}{9}+\frac{(y-3{)}^2}{4}=1$?" (Answer: $(-1,3)$).

• Interpreting a Graph: You will be shown a graph of a conic section and asked to determine its key characteristics from the visual representation.

  • Example: A graph of a hyperbola is shown with its vertices at $(2,5)$ and $(2,-1)$. "What are the coordinates of the center of the hyperbola?" (Answer: The midpoint, $(2,2)$).

Common Mistakes

• Center Coordinate Signs: Incorrectly identifying the center (h, k)$.Thestandardformis$(x-h)^2$, so for a term like $(x+3{)}^2$, the value of $h$ is $-3$, not +3`.

• Forgetting the Square Root: Confusing $a^2,b^2,r^2$ with $a,b,r$. For example, if the equation of a circle is $x^2+y^2=16$, the radius is $r=\sqrt{16}=4$, not 16.

• Mixing up Ellipse and Hyperbola Formulas: Using the focus formula $c^2=a^2-b^2$ for a hyperbola, or $c^2=a^2+b^2$ for an ellipse. Remember, for a hyperbola, $c$ is the longest distance, so you add. For an ellipse, $a$ (the semi-major axis) is the longest distance, so you subtract.

• Incorrect Orientation: Confusing horizontal and vertical orientations for ellipses and hyperbolas. The orientation is determined by which variable's term has the larger denominator (for an ellipse) or which variable's term is positive (for a hyperbola).

• Asymptote Slope: For a hyperbola, incorrectly calculating the slope of the asymptotes. Remember that the slope is always related to "rise over run," which corresponds to the square roots of the numbers under the $y$ and $x$ terms ($\pm a/b$ or $\pm b/a$ depending on orientation).