Identifying Conic Sections

What You’ll Learn

In this lesson you’ll learn how to identify whether an equation represents a circle, parabola, ellipse, or hyperbola, and how to rewrite general equations in standard form.

The Concept

The general second-degree equation is:

$$Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$$

For most Algebra 2 work, we focus on equations where B = 0 (no xy term). You can identify the conic by looking at the squared terms:

What you see	Conic section
x² and y² with same coefficient	Circle
Only one variable squared	Parabola
x² and y² both positive, different coefficients	Ellipse
x² and y² with opposite signs	Hyperbola

Standard form equations:

Circle:

$$(x - h)^2 + (y - k)^2 = r^2$$

Parabola:

$$y = a(x - h)^2 + k \quad\text{or}\quad x = a(y - k)^2 + h$$

Ellipse:

$$\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1$$

Hyperbola:

$$\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1$$

To identify from a general equation:

1. Move all terms to one side so the equation equals zero.
2. Look at which variables are squared and their signs/coefficients.
3. Complete the square if necessary to reach standard form.

Worked Example

1. Circle: x² + y² − 6x + 8y + 9 = 0

Both x² and y² with the same coefficient (1) → circle. Complete the square:

$$\begin{aligned} (x^2 - 6x + 9) + (y^2 + 8y + 16) &= -9 + 9 + 16 \\[1em] (x - 3)^2 + (y + 4)^2 &= 16 \end{aligned}$$

Center (3, −4), radius 4.

2. Parabola: y = 2x² − 8x + 5

Only x is squared → parabola. Convert to vertex form:

$$\begin{aligned} y &= 2(x^2 - 4x) + 5 \\[1em] &= 2(x^2 - 4x + 4 - 4) + 5 \\[1em] &= 2(x - 2)^2 - 3 \end{aligned}$$

Vertex (2, −3), opens upward.

3. Hyperbola: x²/25 − y²/16 = 1

Opposite signs → hyperbola. Already in standard form. Center (0, 0), a = 5, b = 4, horizontal transverse axis.

4. Ellipse: 9x² + 4y² − 36x + 24y + 36 = 0

Both x² and y² positive with different coefficients (9 and 4) → ellipse. Complete the square:

$$\begin{aligned} 9(x^2 - 4x) + 4(y^2 + 6y) &= -36 \\[1em] 9(x^2 - 4x + 4) + 4(y^2 + 6y + 9) &= -36 + 36 + 36 \\[1em] 9(x - 2)^2 + 4(y + 3)^2 &= 36 \\[1em] \frac{(x - 2)^2}{4} + \frac{(y + 3)^2}{9} &= 1 \end{aligned}$$

Center (2, −3), vertical major axis (a = 3 under y²), b = 2.

Real-World Application

Being able to identify conic sections helps in:

• Physics: understanding planetary orbits (ellipses) and projectile paths (parabolas)
• Engineering: designing satellite dishes (parabolas), cooling towers (hyperbolas), and lenses
• Architecture: creating elliptical domes or hyperbolic structures
• Navigation: using hyperbolic navigation systems

Example: astronomers identify elliptical orbits by recognizing the equation form from observational data.

You’ve Got This

The quickest way to identify a conic is to look at the squared terms: same positive coefficients = circle, only one squared = parabola, both positive but different = ellipse, opposite signs = hyperbola. Practice completing the square when the equation is not in standard form. This skill ties together everything you’ve learned about quadratics.

Quiz

The equation $x^2 + y^2 = 49$ represents a:

• A.Ellipse
• B.Circle
• C.Parabola
• D.Hyperbola

Which equation represents a parabola?

• A.$\frac{x^2}{4} + \frac{y^2}{9} = 1$
• B.$x^2 + y^2 = 16$
• C.$\frac{x^2}{9} - \frac{y^2}{4} = 1$
• D.$y = 3x^2 - 6x + 1$

The equation $\frac{x^2}{16} - \frac{y^2}{9} = 1$ is a:

• A.Hyperbola
• B.Ellipse
• C.Circle
• D.Parabola

To identify the type of conic, the most important step is to look at:

• A.The linear terms only
• B.Only the constant term
• C.The signs and coefficients of the squared terms
• D.The degree of the equation

Classify the conic: $4x^2 + 4y^2 - 16x + 8y = 0$.

• A.Ellipse
• B.Circle
• C.Hyperbola
• D.Parabola