Unit 8: Conic Sections

1. Introduction to Conics
2. The Circle
3. The Parabola
4. The Ellipse
5. The Hyperbola
6. Exam Focus Enhancements

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).

Standard Equation: (x - h)² + (y - k)² = r²

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]

Standard Equation: (x²) / (a²) + (y²) / (b²) = 1 (where a > b)

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.

Standard Equation: (x²) / (a²) - (y²) / (b²) = 1

Eccentricity (e): e = √(1 + (b²) / (a²)) (Note: e > 1).
Foci: (± ae, 0).
Transverse Axis: Along x-axis (length 2a).

6. Exam Focus Enhancements

Exam Tips

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.

Common Mistakes

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.

Frequently Asked Questions

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.