General Form of Conic Sections

General Form of Conic Sections#

Circle and Parabola Revision#

Theory#

Before studying the general conic form, let’s review specific conics:

  • Circle: \((x - h)^2 + (y - k)^2 = r^2\) or \(x^2 + y^2 + 2gx + 2fy + c = 0\)

  • Parabola: \(y = ax^2 + bx + c\) or \(x = ay^2 + by + c\)

  • These are special cases of the general second-degree equation

Application#

The equation \(x^2 + y^2 - 4x + 6y - 12 = 0\) represents a circle with:

  • Center: \((2, -3)\)

  • Radius: \(r = \sqrt{4 + 9 + 12} = 5\)

General Form of Second-Degree Equations#

Theory#

The general form of a second-degree equation in two variables is: $\(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)$

where at least one of \(A\), \(B\), or \(C\) is non-zero.

Classification by Discriminant#

The discriminant \(\Delta = B^2 - 4AC\) determines the type of conic:

  1. \(\Delta < 0\): Ellipse (circle if \(A = C\) and \(B = 0\))

  2. \(\Delta = 0\): Parabola

  3. \(\Delta > 0\): Hyperbola

Special Cases#

  1. Circle: \(A = C\), \(B = 0\), and \(\Delta < 0\)

  2. Parabola with vertical axis: \(A \neq 0\), \(B = C = 0\)

  3. Parabola with horizontal axis: \(C \neq 0\), \(A = B = 0\)

  4. Rectangular hyperbola: \(A = -C\), \(B = 0\)

Degenerate Cases#

Sometimes the equation represents:

  • A point (when the conic “collapses”)

  • Two lines (intersecting or parallel)

  • No real points (empty set)

Interactive Visualization: General Conic Explorer#

Application#

Examples#

Example 1: Identifying the Conic Identify the conic represented by \(4x^2 + 4xy + y^2 - 6x - 8y + 9 = 0\).

Solution: Here \(A = 4\), \(B = 4\), \(C = 1\)

Calculate the discriminant: \(\Delta = B^2 - 4AC = 16 - 4(4)(1) = 16 - 16 = 0\)

Since \(\Delta = 0\), this is a parabola.

Example 2: Circle to General Form Convert the circle \((x - 3)^2 + (y + 2)^2 = 16\) to general form.

Solution: Expand: \((x - 3)^2 + (y + 2)^2 = 16\) \(x^2 - 6x + 9 + y^2 + 4y + 4 = 16\) \(x^2 + y^2 - 6x + 4y - 3 = 0\)

In general form: \(A = 1\), \(B = 0\), \(C = 1\), \(D = -6\), \(E = 4\), \(F = -3\)

Verify: \(\Delta = 0 - 4(1)(1) = -4 < 0\) and \(A = C\), \(B = 0\), confirming it’s a circle.

Example 3: Rotating Conic The equation \(x^2 - 2xy + y^2 - 4 = 0\) represents what type of conic?

Solution: Here \(A = 1\), \(B = -2\), \(C = 1\)

\(\Delta = (-2)^2 - 4(1)(1) = 4 - 4 = 0\)

This is a parabola. The presence of the \(xy\) term indicates the parabola is rotated.

Multiple Choice Questions#

Sector Specific Questions: General Form Applications#

Key Takeaways#

Important

General Form of Conics - Essential Concepts

  1. General Equation: \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\)

  2. Classification by Discriminant \(\Delta = B^2 - 4AC\):

    • \(\Delta < 0\): Ellipse (circle if \(A = C\) and \(B = 0\))

    • \(\Delta = 0\): Parabola

    • \(\Delta > 0\): Hyperbola

  3. Key Properties:

    • \(B \neq 0\) indicates rotation

    • Complete the square to find center and axes

    • Degenerate cases possible (point, lines, empty set)

  4. Applications: Satellite dishes, planetary orbits, economics, architecture

  5. Remember: The discriminant quickly identifies the conic type without completing the square

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