Homework 9

Directions:

  1. Show each step of your work and fully simplify each expression.
  2. Turn in your answers in class on a physical piece of paper.
  3. Staple multiple sheets together.
  4. Feel free to use Desmos for graphing.

Answer the following:

  1. For each of the following problems, find an equation of the parabola with vertex at the origin satisfying the following conditions or state it's impossible to.
    1. Focus $(0, 1)$, directrix $x = 1$
    2. Focus $(0, 6)$
    3. Directrix $y = -5$
    4. Focus is on the positive $x$-axis, two units away from the directrix.
    5. This parabola: Hint: you are given a $x$ and $y$ coordinate. Plug them into the proper equation and solve for $p$.
    6. Directrix $y = -2$, focus $(2, 0)$
    7. This parabola:
  2. Write down five different situations in real life where parabolas are used.
  3. What is the geometric definition of an ellipse?
  4. In an ellipse, is the number $a$ ever taken to be bigger than $b$?
  5. For each of the following ellipses:
    • Find the vertices, foci, and length of the minor/major axes.
    • Sketch a rough graph of the ellipse, including the foci and vertices.
    1. $\dfrac{x^2}{25} + \dfrac{y^2}{9} = 1$
    2. $4x^2 + 25y^2 = 100$
    3. $\dfrac{x^2}{18} + \dfrac{2y^2}{81} = 2$
    4. $x^2 = 4 - 2y^2$
    5. $4x^2 + y^2 = 16$
    6. $9x^2 + 4y^2 = 36$
  6. What is the definition of the eccentricity of an ellipse? What does it tell us?
  7. For each of the following problems, find an equation of the ellipse satisfying the following conditions or state it's impossible to.
    1. Eccentricity $2$, foci $(\pm 1, 0)$
    2. Foci $(\pm 4, 0)$, vertices $(0, \pm 5)$
    3. Foci $(\pm 4, 0)$, vertices $(\pm5, 0)$
    4. Foci $(0, \pm 2)$, length of minor axis is $6$
    5. Length of major axis is 10, vertices on the $x$-axis, foci on the $y$-axis
    6. Eccentricity $\dfrac{1}{3}$, foci $(0, \pm 2)$
    7. Vertices $(0, \pm 7)$, Foci $(0, \pm\sqrt{10})$
  8. State three situations in real life where ellipses are used.
  9. What is the geometric definition of a hyperbola?
  10. For each of the following hyperbolas:
    • Find the vertices, foci, length of transverse axis, and asymptotes.
    • Sketch a graph of the hyperbola, including labeling the foci and vertices.
    1. $\dfrac{x^2}{4} - \dfrac{y^2}{16} = 1$
    2. $x^2 - 3y^2 + 12 = 0$
    3. $y^2 - 25x^2 = 100$
    4. $x^2 - 4y^2 - 8 = 0$
    5. $4y^2 - x^2 = 16$
    6. $(y-2x)(y+2x) = 16$