10. Conic Sections
10.1 Introduction
Conic sections are the curves obtained by intersecting a right circular cone with a plane. The main types of conic sections are: Circle, Ellipse, Parabola, and Hyperbola. These curves have important applications in mathematics, physics, engineering, and astronomy.
10.2 Sections of a Cone
The shape of the curve formed depends on the angle at which the plane cuts the cone:
- Circle: Plane cuts perpendicular to the axis of the cone.
- Ellipse: Plane cuts at an angle, but does not intersect the base.
- Parabola: Plane is parallel to a generator (slant edge) of the cone.
- Hyperbola: Plane cuts both nappes (halves) of the cone.
10.3 Circle
Definition: A circle is the set of all points in a plane that are at a fixed 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.
Special Case: Center at origin (0, 0):
x2 + y2 = r2
10.4 Parabola
Definition: A parabola is the set of all points in a plane that are equidistant from a fixed point (focus) and a fixed line (directrix).
Standard Equation:
y2 = 4ax
(Opens right; focus at (a, 0), directrix: x = −a)
General Form:
y = ax2 + bx + c
Vertex: (0, 0) for y2 = 4ax
10.5 Ellipse
Definition: An ellipse is the set of all points in a plane such that the sum of their distances from two fixed points (foci) is constant.
Standard Equation (center at origin, major axis along x-axis):
x2
a2
+
y2
b2
= 1
Where a > b, a = semi-major axis, b = semi-minor axis.
Foci: (±c, 0), where c = √(a2 − b2)
10.6 Hyperbola
Definition: A hyperbola is the set of all points in a plane such that the absolute difference of their distances from two fixed points (foci) is constant.
Standard Equation (center at origin, transverse axis along x-axis):
x2
a2
−
y2
b2
= 1
Where a = semi-transverse axis, b = semi-conjugate axis.
Foci: (±c, 0), where c = √(a2 + b2)
Summary Table: Conic Sections
| Conic | Standard Equation | Key Features |
|---|---|---|
| Circle | (x − h)2 + (y − k)2 = r2 | Center: (h, k), Radius: r |
| Ellipse | x2 a2 + y2 b2 = 1 | Semi-major: a, Semi-minor: b, Foci: (±c, 0) |
| Parabola | y2 = 4ax | Focus: (a, 0), Directrix: x = −a |
| Hyperbola | x2 a2 − y2 b2 = 1 | Semi-transverse: a, Semi-conjugate: b, Foci: (±c, 0) |