Printable practice problems: Ellipse equations general to standard form

📚 Topic Summary

An ellipse is a conic section formed by intersecting a cone with a plane that does not intersect the base. The equation of an ellipse can be expressed in general form, which isn't immediately useful for identifying the ellipse's key features (center, major/minor axes). Converting to standard form allows us to easily read off these properties. The standard form equations are $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ (horizontal major axis) and $\frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1$ (vertical major axis), where $(h, k)$ is the center, $a$ is the semi-major axis, and $b$ is the semi-minor axis.

The process involves completing the square for both $x$ and $y$ terms. Group the $x$ terms together and the $y$ terms together. Factor out the coefficients of the $x^2$ and $y^2$ terms. Complete the square for both $x$ and $y$, being sure to add the same values to both sides of the equation. Finally, divide both sides by the constant term to get the equation in standard form. Let's practice these steps!

🔤 Part A: Vocabulary

Match the following terms with their definitions:

✍️ Part B: Fill in the Blanks

To convert the general form of an ellipse equation to standard form, we must complete the ______ for both $x$ and $y$. This involves grouping the $x$ terms and $y$ terms, then factoring out the leading ______. After completing the square, we divide by a constant to get the equation equal to ______. The standard form reveals the ellipse's ______ and the lengths of its major and minor axes.

🤔 Part C: Critical Thinking

Explain in your own words why converting an ellipse equation from general form to standard form is useful. Provide a specific example of information that is easily obtained from the standard form but not from the general form.