Conic Sections Study Guide

Conic Sections Review

7. Hyperbola Equation Analysis

• Original Equation: $4x^2 - 9y^2 + 16x + 18y + 43 = 0$
• Completing the Square:
  • $4(x^2 + 4x) - 9(y^2 - 2y) = -43$
  • $4(x^2 + 4x + 4) - 9(y^2 - 2y + 1) = -43 + 16 - 9$
  • $4(x + 2)^2 - 9(y - 1)^2 = -36$
• Standard Form:
  • $\frac{(y - 1)^2}{4} - \frac{(x + 2)^2}{9} = 1$
• Type of Conic: Hyperbola
• Center: $(-2, 1)$
• Vertices: $(-2, 3), (-2, -1)$
• Foci:
  • $c^2 = b^2 + a^2 = 9 + 4 = 13$
  • $c = \sqrt{13}$
  • Foci: $(-2, 1 \pm \sqrt{13})$
• Directrices: Not explicitly calculated but would be horizontal lines.
• Equations of Asymptotes: $y - 1 = \pm \frac{2}{3}(x + 2)$
• Change in Conic:
  • If the equation was $4x^2 - 9y^2 + 16x - 18y + 43 = 0$, the sign of the $y$ term changes.
  • $4(x^2 + 4x + 4) - 9(y^2 - 2y + 1) = -43 + 16 - 9$
  • $4(x + 2)^2 - 9(y - 1)^2 = -36$
  • $\frac{(x + 2)^2}{9} - \frac{(y - 1)^2}{4} = -1$

8. Finding Standard Equation of a Conic

• Foci: $(5, -2)$ and $(5, 8)$
• Vertices: $(5, 2)$ and $(5, 4)$
• Center: Midpoint of the vertices or foci, which is $(5, 3)$
• Distance between vertices (2a): $4 - 2 = 2$, so $a = 1$
• Distance between foci (2c): $8 - (-2) = 10$, so $c = 5$
• Since $c^2 = a^2 + b^2$, then $b^2 = c^2 - a^2 = 25 - 1 = 24$
• The major axis is vertical because the foci and vertices have the same x-coordinate.
• Standard Equation: $\frac{(y - 3)^2}{1} - \frac{(x - 5)^2}{24} = 1$

9. Thunder Location Problem

• Distance between you and your friend: Half a mile, or 2640 feet.
• Speed of sound: 1100 feet/second.
• Time difference in hearing thunder: 1.5 seconds.
• Distance difference: $1.5 \times 1100 = 1650$ feet.
• Let your location be $(-1320, 0)$ and your friend's location be $(1320, 0)$.
• The equation representing possible locations of the thunder is a hyperbola:
  • $|d<em>1 - d</em>2| = 1650$, where $d<em>1$ and $d</em>2$ are the distances from the thunder to you and your friend, respectively.
  • $2a = 1650$, so $a = 825$
  • The foci are at $(\pm 1320, 0)$, so $c = 1320$
  • $b^2 = c^2 - a^2 = 1320^2 - 825^2$
  • $b^2 = 1742400 - 680625 = 1061775$
• Equation: $\frac{x^2}{825^2} - \frac{y^2}{1061775} = 1$
  • $\frac{x^2}{680625} - \frac{y^2}{1061775} = 1$

Honors Algebra II/Trig - Unit 9 Review #2: Conics

1. Conditions for Equations

• Line: $A = 0, C = 0, B \neq 0$ and/or $D \neq 0$
• Hyperbola: A/C < 0, A \neq 0, C \neq 0
• Parabola: $A = 0, C \neq 0$ or $C = 0, A \neq 0$
• Ellipse: $A \neq 0, C \neq 0$, A & C have the same sign.
• Circle: $A = C$

2. Comparing and Contrasting Conic Sections

• Conic 1: $\frac{(x-1)^2}{12} + \frac{(y+3)^2}{19} = 1$
  • Type: Ellipse
  • Center: $(1, -3)$
  • x-radius: $\sqrt{12}$
  • y-radius: $\sqrt{19}$
  • Focal length: $c^2 = b^2 - a^2 = 19 - 12 = 7$, so $c = \sqrt{7}$
  • Foci: $(1, -3 \pm \sqrt{7})$
• Conic 2: $\frac{(x-1)^2}{12} - \frac{(y+3)^2}{19} = 1$
  • Type: Hyperbola, opens in x-direction
  • Center: $(1, -3)$
  • Focal Length: $c^2 = a^2 + b^2 = 12 + 19 = 31$, so $c = \sqrt{31}$
  • Foci: $(1 \pm \sqrt{31}, -3)$
• Conic 3: $\frac{(y+3)^2}{19} - \frac{(x-1)^2}{12} = 1$
  • Type: Hyperbola, opens in y-direction
  • Center: $(1, -3)$
  • Focal Length: $c^2 = a^2 + b^2 = 12 + 19 = 31$, so $c = \sqrt{31}$
  • Foci: $(1, -3 \pm \sqrt{31})$
  • Asymptotes: $y + 3 = \pm \sqrt{\frac{19}{12}} (x - 1)$
• Similarities:
  • All have the same center $(1, -3)$

3. Identifying Conic Type and Equation in General Form

• Condition: For each point, its distance from the point $(-4, 7)$ is its distance from the point $(5, -1)$.
  • $\sqrt{(x+4)^2 + (y-7)^2} = \sqrt{(x-5)^2 + (y+1)^2}$
  • $(x+4)^2 + (y-7)^2 = (x-5)^2 + (y+1)^2$
  • $x^2 + 8x + 16 + y^2 - 14y + 49 = x^2 - 10x + 25 + y^2 + 2y + 1$
  • $18x - 16y + 65 = 26$
  • $18x - 16y + 39 = 0$
  • Type: Line

4. Solving System of Equations

• System:
  • $4x^2 + y^2 = 36$
  • $y = 2x - 1$
• Substitute $y$ in the first equation:
  • $4x^2 + (2x - 1)^2 = 36$
  • $4x^2 + 4x^2 - 4x + 1 = 36$
  • $8x^2 - 4x - 35 = 0$
• This does not appear to factor nicely, using the original equation y = -2x-1 from the image
  • $4x^2 + (-2x-1)^2 = 36$
  • $4x^2 + 4x^2 + 4x + 1 = 36$
  • $8x^2 + 4x - 35 = 0$

5. Sketching a System with Three Solutions

• Conics:
  • $Ax^2 + By^2 = C$ (Ellipse)
  • $Dy^2 + Ey + Fx = G$ (Parabola)
• Visualization: The ellipse and parabola should intersect at exactly three points.

6. Conic Information

• Equation: $5y^2 - 40y + 5x^2 + 67 = 0$
  • $5x^2 + 5(y^2 - 8y) = -67$
  • $5x^2 + 5(y^2 - 8y + 16) = -67 + 80$
  • $5x^2 + 5(y - 4)^2 = 13$
  • $x^2 + (y - 4)^2 = \frac{13}{5}$
• Type of Conic: Circle
• Center: $(0, 4)$
• Changed Equation: $5y^2 - 40y - 5x^2 + 67 = 0$
  • The conic would become a hyperbola.