Unit 8: Conic Sections
Contents
1. Introduction to Conics
Conic sections are the curves obtained by intersecting a right circular cone with a plane. Depending on the angle of the plane, we get four distinct types of curves: Circle, Ellipse, Parabola, and Hyperbola.
[Image of conic sections: circle, ellipse, parabola, hyperbola]2. The Circle
A circle is the set of all points in a plane that are at a constant distance (radius) from a fixed point (center).
Where (h, k) is the center and r is the radius. If the center is at origin (0,0), the equation simplifies to x2 + y2 = r2.
3. The Parabola
A parabola is the set of all points in a plane that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.
| Standard Form | Opens Towards | Focus | Directrix |
|---|---|---|---|
| y2 = 4ax | Right | (a, 0) | x = -a |
| y2 = -4ax | Left | (-a, 0) | x = a |
| x2 = 4ay | Upward | (0, a) | y = -a |
| x2 = -4ay | Downward | (0, -a) | y = a |
4. The Ellipse
An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points (foci) is constant.
[Image of an ellipse showing major axis, minor axis, and foci]- Foci: (± ae, 0) where e is eccentricity.
- Eccentricity (e): e = √(1 - (b2) / (a2))
- Length of Latus Rectum: (2b2) / (a)
5. The Hyperbola
A hyperbola is the set of all points in a plane, the difference of whose distances from two fixed points (foci) is a positive constant.
- Eccentricity (e): e = √(1 + (b2) / (a2)) (Note: e > 1).
- Foci: (± ae, 0).
- Transverse Axis: Along x-axis (length 2a).
6. Exam Focus Enhancements
- Eccentricity Comparison: Remember the values of e: Circle (e=0), Ellipse (0 < e < 1), Parabola (e=1), Hyperbola (e > 1).
- The Sum/Difference Rule: If the problem says "Sum of distances is constant," it's an Ellipse. If it says "Difference of distances is constant," it's a Hyperbola.
- Standard Form: Always convert your given equation into standard form (where the right side is 1 for Ellipse/Hyperbola) before identifying a and b.
- Hyperbola Signs: Confusing (x2) / (a2) - (y2) / (b2) = 1 (horizontal) with (y2) / (a2) - (x2) / (b2) = 1 (vertical). Look at which variable has the positive sign!
- Radius Squared: In circles, x2 + y2 = 25 means the radius is 5, not 25.
- a and b for Ellipse: Forgetting that a is always the semi-major axis length. If the denominator under y2 is larger, the ellipse is vertical.
Q: What is the Latus Rectum?
A: It is a chord passing through the focus and perpendicular to the major (or transverse) axis.
Q: Can eccentricity be negative?
A: No, eccentricity is a ratio of distances and is always non-negative.