The Main Idea

The general second-degree equation for conic sections is [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[/latex].

Depending on the coefficients, this equation can represent a circle, ellipse, parabola, or hyperbola. These are called nondegenerate conics. Identifying which conic you have involves looking at the coefficients [latex]A,B,C[/latex].

Quick Tips: How to Identify the Conic

1. Check the Cross-Term ([latex]Bxy[/latex])

   • If [latex]B\ne 0[/latex], the conic is rotated. Identification requires more advanced analysis (rotation of axes).

   • If [latex]B=0[/latex], proceed with simpler tests below.

2. Circle

   • [latex]A=C \ne 0[/latex].

   • Example: [latex]x^{2}+y^{2}-16=0[/latex].

3. Ellipse (not circle)

   • [latex]A \ne C[/latex], both positive.

   • Example: [latex]\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1[/latex].

4. Parabola

   • One squared term only (either [latex]A=0[/latex] or [latex]C=0[/latex]).

   • Example: [latex]y^{2}=8x[/latex].

5. Hyperbola

   • [latex]A[/latex] and [latex]C[/latex] have opposite signs.

   • Example: [latex]\dfrac{x^{2}}{9}-\dfrac{y^{2}}{4}=1[/latex].

The Main Idea

When the general conic equation has a cross-product term [latex]Bxy[/latex], the conic is rotated relative to the coordinate axes. To write its equation in standard form, we apply a rotation of axes. This change of variables removes the [latex]xy[/latex]-term so the conic can be recognized (circle, ellipse, parabola, or hyperbola).

The rotation formulas are:

• [latex]x = x'\cos\theta - y'\sin\theta[/latex]

• [latex]y = x'\sin\theta + y'\cos\theta[/latex]

The angle [latex]\theta[/latex] that eliminates the [latex]xy[/latex]-term satisfies:

[latex]\tan(2\theta) = \dfrac{B}{A-C}[/latex].

Quick Tips: Writing Rotated Conics in Standard Form

1. Identify the Cross Term

   • If [latex]B=0[/latex], no rotation is needed.

   • If [latex]B\ne 0[/latex], compute [latex]\theta[/latex] using [latex]\tan(2\theta)=\dfrac{B}{A-C}[/latex].

2. Apply Rotation Formulas

   • Substitute [latex]x, y[/latex] in terms of [latex]x', y'[/latex].

   • Expand and simplify to eliminate the [latex]x'y'[/latex]-term.

3. Recognize the Standard Form

   • Once simplified, the equation will match a conic standard form:

     • Ellipse: [latex]\dfrac{x'^{2}}{a^{2}}+\dfrac{y'^{2}}{b^{2}}=1[/latex]

     • Hyperbola: [latex]\dfrac{x'^{2}}{a^{2}}-\dfrac{y'^{2}}{b^{2}}=1[/latex]

     • Parabola: [latex]y'^{2}=4px'[/latex] or [latex]x'^{2}=4py'[/latex]

The Main Idea

Even if a conic equation includes a cross-product term [latex]Bxy[/latex], it is possible to identify the type of conic without actually rotating the axes. This is done using the discriminant: Δ=B2−4AC where the general conic equation is [latex]Ax^{2} + Bxy + Cy^{2} + Dx + Ey + F = 0[/latex].

The discriminant reveals the type of conic:

• [latex]\Delta < 0[/latex] → ellipse (if [latex]A=C[/latex] and [latex]B=0[/latex], it’s a circle).

• [latex]\Delta = 0[/latex] → parabola.

• [latex]\Delta > 0[/latex] → hyperbola.

Quick Tips: Identifying Conics

1. Write in General Form

   • Ensure the equation is in [latex]Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0[/latex].

2. Compute the Discriminant

   • [latex]\Delta = B^{2}-4AC[/latex].

3. Interpret the Result

   • If [latex]\Delta < 0[/latex] → ellipse.

   • If [latex]\Delta = 0[/latex] → parabola.

   • If [latex]\Delta > 0[/latex] → hyperbola.

4. Special Case: Circle

   • If [latex]\Delta < 0[/latex] and [latex]A=C[/latex], with [latex]B=0[/latex], the conic is a circle.