Chapter 10-11 Notes: Conic Sections

Welcome back to Professor Baker's Math Class! This post summarizes the key concepts we covered in our recent sessions on conic sections, focusing on circles, ellipses, and hyperbolas. We explored how to graph these shapes, write their equations, and identify their crucial features.

Here's a breakdown of the topics we discussed:

Circles

  • Graphing Circles: Understanding the standard and general forms of a circle's equation is key to graphing.
    • Standard Form: $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.
    • General Form: $Ax^2 + Ay^2 + Dx + Ey + F = 0$. Completing the square allows us to convert from general to standard form.
  • Writing Equations of Circles: We learned how to write the equation of a circle given various pieces of information.
    • Center and Radius: Directly apply the standard form.
    • Center and a Point on the Circle: Use the distance formula to find the radius.
    • Endpoints of a Diameter: Find the midpoint (center) and the distance from the center to an endpoint (radius).

Ellipses

  • Graphing Ellipses: We focused on graphing ellipses from both standard and general forms.
    • Standard Form (centered at origin): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Here, $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.
    • General Form: $Ax^2 + By^2 = C$ (where A and B are positive and unequal). As with circles, we can manipulate this form to find the standard equation.
  • Key Features of Ellipses: It's important to identify the center, vertices, and foci of an ellipse. Remember the relationship: $c^2 = a^2 - b^2$, where $c$ is the distance from the center to each focus.
  • Writing Equations of Ellipses: We covered how to write the equation of an ellipse given different sets of information.
    • Center, an Endpoint of an Axis, and the Length of the Other Axis: Use this information to determine $a$ and $b$, and then plug into the standard form.
    • Foci and the Major Axis Length: Use the distance from the center to the foci ($c$) and half of the major axis length ($a$) to find $b$ using the relationship $c^2 = a^2 - b^2$.

Hyperbolas

  • Graphing Hyperbolas: Graphing hyperbolas requires understanding the standard form and identifying the asymptotes.
    • Standard Form (centered at origin): $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (opens horizontally) or $\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ (opens vertically).
    • General Form: $Ax^2 - By^2 = C$ or $-Ax^2 + By^2 = C$ (A and B are positive).
  • Key Features of Hyperbolas: Focus on finding the center, vertices, foci, and asymptotes. Remember the relationship: $c^2 = a^2 + b^2$.
  • Writing Equations of Hyperbolas: We practiced writing the equation of a hyperbola when given the foci and vertices. The distance between the vertices is $2a$, and the distance between the foci is $2c$. Use these values and the relationship $c^2 = a^2 + b^2$ to solve for $a$ and $b$ and write the equation in standard form.

Keep practicing these concepts, and you'll master conic sections in no time! Don't forget to review the video and attached PDFs for more detailed explanations and examples. Good luck with your studies!