DMA - 9. Circle, Ellipse and Hyperbola

DMA

CHAPTER - 9

CONIC SECTIONS (PARABOLA, ELLIPSE AND HYPERBOLA)

Learning Outcomes

Understand Basic Concepts of Conic Sections

Define conic sections and explain how they are formed by cutting a cone.
Identify and describe the properties of parabolas, ellipses, and hyperbolas.
Recognize real-life examples of each conic section in maritime operations and navigation.

Parabola

Understand the standard equation of a parabola and identify its key features (focus, directrix, axis, and vertex).
Solve simple problems involving parabolic paths, such as the motion of projectiles (e.g., launching flares).
Apply the properties of parabolas in understanding radar and satellite communication paths.

Ellipse

Understand the standard equation of an ellipse and identify its key features (major axis, minor axis, foci).
Apply the concept of ellipses to understand satellite orbits and positioning systems.
Solve problems involving the measurement of distances using elliptical paths.

Hyperbola

Recognize the standard equation of a hyperbola and identify its key features (foci, asymptotes, transverse and conjugate axes).
Understand the application of hyperbolas in radio navigation systems (like LORAN) to determine position.
Solve simple problems involving hyperbolic paths in navigation.

Apply Conic Sections in Maritime Contexts

Use the properties of conic sections to solve problems involving radar range, satellite tracking, and communication paths.
Apply mathematical understanding to interpret data involving distances and angles using conic concepts.

Interpret and Communicate Results

Interpret graphical data related to conic sections in navigational tools and systems.
Clearly explain how conic concepts apply to real-world maritime operations.

The word conic is derived from the word cone. A cone can be generated as follows. Let a and g be two distinct lines that intersect at a point V. Rotate the line g around the line , maintaining the angle between a and g. The resulting surface is called the cone. The line is called the axis of the cone; the line g is called a generator of the cone. The point V where the axis and generator intersect is called the vertex of the cone.

The curves that result by intersecting a cone with a plane are called conic sections or, simply conics.

The most interesting conics occur when the intersection plane does not pass through the vertex. These conics are circles, ellipses, hyperbolas and parabolas.

9.1 Equation of A Conic-General Form

Let us assume the focus of a conic at (m, n) and that the directrix is the line x + by a+ c = 0.

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