Training Conic Sections Classifying Conics & Applications
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Classifying Conics & Applications

24 min Conic Sections
Given the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant B² − 4AC classifies the conic: negative gives an ellipse (or circle), zero gives a parabola, and positive gives a hyperbola. When B ≠ 0 the axes are rotated, and a rotation of coordinates through the angle θ = ½ arctan(B/(A − C)) eliminates the xy term. In applied settings, conic sections describe satellite orbits, optical reflectors, bridge arches, and antenna cross-sections. Choosing the right conic for a given design problem—parabolic for focusing, elliptical for distributing, hyperbolic for cooling—is a key skill in engineering and physics.

Classifying Conics & Applications

General Second-Degree Equation

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

When $B = 0$:

  • $A = C$: Circle
  • $A$ or $C = 0$ (but not both): Parabola
  • $A$ and $C$ same sign, $A \neq C$: Ellipse
  • $A$ and $C$ opposite signs: Hyperbola
Discriminant Test (when $B \neq 0$)

$$\Delta = B^2 - 4AC$$

  • $\Delta < 0$: Ellipse (or circle)
  • $\Delta = 0$: Parabola
  • $\Delta > 0$: Hyperbola
Example 1

Classify: $3x^2 + 3y^2 - 12x + 6y - 9 = 0$.

  1. $A = C = 3$ → Circle.
  2. Divide by 3 and complete the square.
Example 2

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

  1. $A = 2$, $C = -1$.
  2. Opposite signs → Hyperbola.
Example 3

A bridge arch is semi-elliptical: 40 m wide, 10 m high at center. Find the height 12 m from center.

  1. Ellipse: $\frac{x^2}{400}+\frac{y^2}{100}=1$.
  2. At $x=12$:
  3. $y^2 = 100(1-144/400) = 64$, $y = 8$ m.

Practice Problems

1. Classify: $x^2 + 4y^2 = 16$.
2. Classify: $y = x^2 + 3x + 1$.
3. Classify: $x^2 - y^2 = 4$.
4. Classify: $x^2 + y^2 = 25$.
5. Use the discriminant: $x^2 + 4xy + 4y^2 = 1$.
6. A whispering gallery is elliptical: $a = 50$ ft, $b = 30$ ft. How far apart are the foci?
7. Classify: $9x^2 + 4y^2 - 36x + 8y = -4$.
8. Classify: $y^2 - 4x + 6y + 1 = 0$.
9. Classify: $4x^2 + 4y^2 + 8x - 16y + 4 = 0$.
10. What conic has $e = 1$?
11. What conic has $0 < e < 1$?
12. A headlight reflector is parabolic, 6 in wide and 4 in deep. Find the focus.
Show Answer Key

1. Ellipse ($A=1$, $C=4$, same sign, different)

2. Parabola (only $x^2$ term)

3. Hyperbola (opposite signs)

4. Circle ($A = C = 1$)

5. $\Delta = 16 - 4(1)(4) = 0$ → Parabola

6. $c = \sqrt{2500-900} = 40$; foci $80$ ft apart

7. Ellipse ($A=9$, $C=4$, same sign)

8. Parabola (no $x^2$ term)

9. Circle ($A = C = 4$)

10. Parabola

11. Ellipse

12. $x^2 = 4py$; $3^2 = 4p(4)$; $p = 9/16 \approx 0.56$ in

🔍 Conic Classifier
Discriminant B²−4AC
Classification
Rotation angle θ
Hyperbolas Placement Test Practice — Conic Sections