Precalculus Analytic Geometry and Polar Coordinates Syllabus

Analytic Geometry and Polar Coordinates syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Analytic Geometry and Polar Coordinates in Precalculus, not a summary of it.

1. 1

   Conic Sections

   3 topics

   • Parabolas
   • Ellipses
   • Hyperbolas
2. 2

   Translations and Classification of Conics

   2 topics

   • Conics with Shifted Centers
   • Identifying Conics by Equation
3. 3

   Parametric Equations

   3 topics

   • Plotting Parametric Curves
   • Eliminating the Parameter
   • Projectile Motion Models
4. 4

   Polar Coordinates

   3 topics

   • The Polar Coordinate System
   • Converting Between Polar and Rectangular
   • Graphs of Polar Equations

Analytic Geometry and Polar Coordinates flashcards for Precalculus

24 of 50 cards from the Analytic Geometry and Polar Coordinates deck — real questions with worked answers.

1. What is the geometric definition of a parabola?

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

2. For a parabola opening up or down with vertex at the origin, give the standard equation and the location of the focus and directrix.

   $x^{2} = 4py$. The focus is at $(0, p)$ and the directrix is the line $y = -p$.

3. For a parabola opening left or right with vertex at the origin, give the standard equation and the location of the focus and directrix.

   $y^{2} = 4px$. The focus is at $(p, 0)$ and the directrix is the line $x = -p$.

4. In the parabola equation $x^{2} = 4py$, what does the sign of $p$ tell you?

   If $p > 0$ the parabola opens upward; if $p < 0$ it opens downward. (For $y^{2}=4px$: $p>0$ opens right, $p<0$ opens left.)

5. What is the latus rectum (focal width) of a parabola, and how long is it?

   The latus rectum is the chord through the focus perpendicular to the axis of symmetry. Its length is $|4p|$.

6. What is the geometric definition of an ellipse?

   An ellipse is the set of all points the sum of whose distances from two fixed points (the foci) is a constant, equal to $2a$.

7. Give the standard equation of an ellipse centered at the origin with a horizontal major axis.

   $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with $a > b > 0$; the major axis lies along the $x$-axis.

8. For an ellipse, how are $a$, $b$, and $c$ (the focal distance) related?

   $c^{2} = a^{2} - b^{2}$, where $a$ is the semi-major axis, $b$ the semi-minor axis, and $c$ the distance from center to each focus.

9. For an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ with $a>b$, where are the vertices and foci located?

   Vertices at $(\pm a, 0)$, co-vertices at $(0, \pm b)$, and foci at $(\pm c, 0)$ where $c = \sqrt{a^{2}-b^{2}}$.

10. How do you determine whether the major axis of an ellipse is horizontal or vertical from its equation?

    The major axis lies along the axis of the variable with the larger denominator ($a^{2}$). If $a^{2}$ is under $x^{2}$ it is horizontal; if under $y^{2}$ it is vertical.

11. Define the eccentricity of an ellipse and give its range of values.

    $e = \frac{c}{a}$, with $0 < e < 1$. Values near $0$ give a nearly circular ellipse; values near $1$ give a very elongated one.

12. What is the geometric definition of a hyperbola?

    A hyperbola is the set of all points the absolute difference of whose distances from two fixed points (the foci) is a constant, equal to $2a$.

13. Give the standard equation of a hyperbola centered at the origin opening left/right.

    $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$, with vertices at $(\pm a, 0)$ and transverse axis along the $x$-axis.

14. Give the standard equation of a hyperbola centered at the origin opening up/down.

    $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$, with vertices at $(0, \pm a)$ and transverse axis along the $y$-axis.

15. For a hyperbola, how are $a$, $b$, and $c$ related?

    $c^{2} = a^{2} + b^{2}$, where $c$ is the distance from center to each focus and $a$ is the distance from center to each vertex.

16. What are the asymptotes of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$?

    $y = \pm \frac{b}{a}x$.

17. What are the asymptotes of the hyperbola $\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1$?

    $y = \pm \frac{a}{b}x$.

18. How do you tell which way a hyperbola opens from its standard equation?

    It opens toward the axis of the positive term: if the $x^{2}$ term is positive it opens left/right; if the $y^{2}$ term is positive it opens up/down. (Unlike an ellipse, this is not about which denominator is larger.)

19. How do the standard-form equations of a conic change when the center/vertex is shifted from the origin to $(h, k)$?

    Replace $x$ with $(x-h)$ and $y$ with $(y-k)$. For example an ellipse becomes $\frac{(x-h)^{2}}{a^{2}} + \frac{(y-k)^{2}}{b^{2}} = 1$.

20. Write the standard equation of a parabola with vertex $(h,k)$ opening up or down.

    $(x-h)^{2} = 4p(y-k)$, with focus at $(h, k+p)$ and directrix $y = k - p$.

21. What algebraic technique converts a general conic equation into standard (shifted) form?

    Completing the square in $x$ and/or $y$ to produce perfect-square terms $(x-h)^{2}$ and $(y-k)^{2}$.

22. For the general second-degree equation $Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$ with $B = 0$, how do you classify a parabola?

    It is a parabola when exactly one of $A$ or $C$ is zero (i.e., $A \cdot C = 0$ but not both), so only one squared term appears.

23. For $Ax^{2} + Cy^{2} + Dx + Ey + F = 0$ (with $B=0$), how do you tell an ellipse from a hyperbola?

    If $A$ and $C$ have the same sign (and $A \neq C$), it is an ellipse (a circle if $A = C$); if $A$ and $C$ have opposite signs, it is a hyperbola.

24. What is the discriminant test for classifying a general conic $Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0$?

    Compute $B^{2} - 4AC$: if $< 0$ it is an ellipse (or circle), if $= 0$ a parabola, if $> 0$ a hyperbola.

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