5.3: Conic Sections | Geometry Honors | Code Scholars Tutoring

How it works

Standard forms distinguish circles, ellipses, parabolas, and hyperbolas. For an ellipse, the larger denominator gives the major axis. For a hyperbola, the positive squared term gives the transverse-axis direction.

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\quad(a>b>0)

Worked examples

Example 1

Identify $\displaystyle x^2/25+y^2/9=1$ and find its vertices.

It is an ellipse centered at the origin with horizontal major axis. Its major vertices are $\displaystyle (\pm5,0)$ .

Example 2

For $\displaystyle y^2=8x$ , find the focus and directrix.

Compare $\displaystyle y^2=4px$ : $\displaystyle p=2$ . The focus is $\displaystyle (2,0)$ and directrix is $\displaystyle x=-2$ .

Problems

Problem 5.3.1

For $\displaystyle x^2/25+y^2/9=1$ , find the foci.

Ellipse with horizontal semiaxis 5 and vertical semiaxis 3. — The major axis is horizontal because 25 is the larger denominator.

Show worked solution

$\displaystyle c^2=a^2-b^2=25-9=16$ , so the foci are $\displaystyle (\pm4,0)$ .

Ellipse with horizontal semiaxis 5 and vertical semiaxis 3. The solution adds labeled relationships. — The focal distance is sqrt(25 - 9) = 4.

Problem 5.3.2

Find the vertices and asymptotes of $\displaystyle x^2/9-y^2/4=1$ .

Show worked solution

Vertices: $\displaystyle (\pm3,0)$ . Asymptotes: $\displaystyle y=\pm(2/3)x$ .

Problem 5.3.3

Explain why $\displaystyle x^2+y^2=-1$ has no real graph.

Show worked solution

Each square is nonnegative, so their sum cannot equal $\displaystyle -1$ .

Lesson Problems PDF Solutions PDF Unit MCQ

How it works

Worked examples

Example 1

Example 2

Problems

Problem 5.3.1

Problem 5.3.2

Problem 5.3.3

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