Identifying Conic Sections

Now that you know the standard forms for circles, parabolas, ellipses, and hyperbolas, the next skill is identifying which conic you are looking at when the equation is given in general form. In this lesson you will learn to classify conics using the discriminant, convert general-form equations to standard form, and recognize degenerate cases.

The General Second-Degree Equation

Every conic section can be written in the form:

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

where $A$, $B$, $C$, $D$, $E$, $F$ are real constants and at least one of $A$, $B$, $C$ is nonzero. In this course we focus on the case where $B = 0$ (no $xy$ term), which covers all conics aligned with the coordinate axes. When $B \neq 0$, the conic is rotated — a topic for precalculus.

Classification When B = 0

When there is no $xy$ term, classification depends on the coefficients of $x^2$ and $y^2$:

Memory aid: Think of it as comparing the coefficients of $x^2$ and $y^2$:

• Equal coefficients = circle (perfect symmetry)
• Same sign, different coefficients = ellipse (stretched circle)
• Opposite signs = hyperbola (one term is subtracted)
• One is zero = parabola (only one squared term)

Quick Examples

• $3x^2 + 3y^2 - 12x + 6y - 9 = 0$: $A = C = 3$ (equal) → Circle
• $4x^2 + 9y^2 - 16x + 54y + 61 = 0$: $A = 4$, $C = 9$ (same sign, different) → Ellipse
• $4x^2 - 9y^2 - 24x - 36y - 36 = 0$: $A = 4$, $C = -9$ (opposite signs) → Hyperbola
• $x^2 + 6x - 4y + 5 = 0$: $A = 1$, $C = 0$ → Parabola

The Discriminant Method (General Case)

When $B \neq 0$, you need the discriminant:

$\Delta = B^2 - 4AC$

When $B = 0$, this simplifies: $\Delta = -4AC$, and the sign of $\Delta$ depends entirely on whether $A$ and $C$ have the same sign (negative $\Delta$, ellipse), opposite signs (positive $\Delta$, hyperbola), or one is zero ($\Delta = 0$, parabola).

Converting to Standard Form: The Complete Process

The general method for any conic (with $B = 0$):

1. Group the $x$ terms and $y$ terms
2. Factor out the leading coefficients from each group
3. Complete the square for each variable
4. Simplify and divide to get the appropriate standard form

Worked Example 1: Is It an Ellipse or a Circle?

Classify and convert $2x^2 + 2y^2 - 8x + 12y + 2 = 0$.

Classify: $A = 2$, $C = 2$. Equal coefficients → Circle.

Convert:

$2(x^2 - 4x) + 2(y^2 + 6y) = -2$

$2(x^2 - 4x + 4) + 2(y^2 + 6y + 9) = -2 + 8 + 18$

$2(x - 2)^2 + 2(y + 3)^2 = 24$

$(x - 2)^2 + (y + 3)^2 = 12$

Center: $(2, -3)$, radius: $r = \sqrt{12} = 2\sqrt{3}$.

Worked Example 2: Identifying a Parabola

Classify and convert $y^2 + 8x - 6y + 1 = 0$.

Classify: $A = 0$ (no $x^2$ term), $C = 1$. One coefficient is zero → Parabola.

Convert:

$y^2 - 6y = -8x - 1$

$y^2 - 6y + 9 = -8x - 1 + 9$

$(y - 3)^2 = -8x + 8 = -8(x - 1)$

This is $(y - k)^2 = 4p(x - h)$ with vertex $(1, 3)$, $4p = -8$, $p = -2$. Opens left.

Worked Example 3: Classifying Without Converting

Classify $5x^2 - 3y^2 + 10x + 12y - 22 = 0$.

$A = 5$, $C = -3$. Opposite signs → Hyperbola. (If you needed full details, you would complete the square.)

Worked Example 4: A Tricky General Form

Classify and convert $9x^2 + 4y^2 + 54x - 8y + 49 = 0$.

Classify: $A = 9$, $C = 4$. Same sign, different → Ellipse.

Convert:

$9(x^2 + 6x) + 4(y^2 - 2y) = -49$

$9(x^2 + 6x + 9) + 4(y^2 - 2y + 1) = -49 + 81 + 4$

$9(x + 3)^2 + 4(y - 1)^2 = 36$

$\frac{(x + 3)^2}{4} + \frac{(y - 1)^2}{9} = 1$

Vertical major axis. Center $(-3, 1)$, $a = 3$, $b = 2$.

Degenerate Conics

Sometimes the general equation does not produce a “real” conic. These degenerate cases occur when the equation factors or simplifies to something unexpected:

Example: Degenerate Hyperbola

Classify $x^2 - 4y^2 = 0$.

$A = 1$, $C = -4$. Opposite signs → should be a hyperbola. But:

$x^2 - 4y^2 = (x - 2y)(x + 2y) = 0$

This gives two lines: $x = 2y$ and $x = -2y$. The “hyperbola” has degenerated into its asymptotes.

Example: Degenerate Ellipse

After completing the square: $(x - 1)^2 + (y + 2)^2 = 0$

The only solution is the single point $(1, -2)$.

A Systematic Decision Flowchart

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

1. Is $B = 0$? (For this course, usually yes.)
2. Count the squared terms:
   • Only $x^2$: Parabola (vertical axis, solve for $y$)
   • Only $y^2$: Parabola (horizontal axis, solve for $x$)
   • Both $x^2$ and $y^2$: proceed to step 3
3. Compare coefficients of $x^2$ and $y^2$:
   • Equal → Circle (complete the square, watch for degenerates)
   • Same sign, different → Ellipse (complete the square)
   • Opposite signs → Hyperbola (complete the square)
4. Complete the square and check for degenerates

Real-World Application: Identifying Curves in Engineering

An engineer encounters the equation $16x^2 + 25y^2 - 160x - 200y + 400 = 0$ describing the cross-section of a tunnel.

Step 1: $A = 16$, $C = 25$. Same sign, different → Ellipse.

Step 2: Convert:

$16(x^2 - 10x) + 25(y^2 - 8y) = -400$

$16(x^2 - 10x + 25) + 25(y^2 - 8y + 16) = -400 + 400 + 400$

$16(x - 5)^2 + 25(y - 4)^2 = 400$

$\frac{(x - 5)^2}{25} + \frac{(y - 4)^2}{16} = 1$

The tunnel cross-section is an ellipse centered at $(5, 4)$ with horizontal semi-axis 5 and vertical semi-axis 4. The engineer can use this to calculate clearance heights at any horizontal position.

Common Mistakes

1. Forgetting to check for degenerate cases. Always verify that the right side is positive after completing the square for ellipses and circles.
2. Confusing the discriminant formula. It is $B^2 - 4AC$, not $B^2 - 4AC$ from the quadratic formula (same form, different context).
3. Not factoring out coefficients before completing the square. If the equation has $4x^2 - 8x$, factor to $4(x^2 - 2x)$ before completing the square inside.
4. Misidentifying when coefficients differ only by sign versus value. $3x^2 + 3y^2$ is a circle, $3x^2 + 5y^2$ is an ellipse, $3x^2 - 5y^2$ is a hyperbola.

Practice Problems

Key Takeaways

• The general second-degree equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ describes all conic sections
• When $B = 0$: compare $A$ and $C$ — equal gives a circle, same-sign-different gives an ellipse, opposite signs give a hyperbola, one zero gives a parabola
• The discriminant $B^2 - 4AC$ classifies conics in the general case: negative = ellipse, zero = parabola, positive = hyperbola
• Always complete the square and check for degenerate cases (single point, no graph, or intersecting lines)
• This classification skill ties together everything from the conic sections unit

Return to College Algebra for more topics in this section.