Conics.

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Presentation transcript:

Conics

Conic Sections (1) Circle A circle is formed when i.e. when the plane  is perpendicular to the axis of the cones.

Equation of a circle We should recall that the equation of a circle is given by: What is the equation of this circle?

Conic Sections (2) Ellipse An ellipse is formed when i.e. when the plane  cuts only one of the cones, but is neither perpendicular to the axis nor parallel to the a generator.

Equation of an ellipse What transformation would change the circle into the ellipse? What is the equation of the ellipse? Can you write it another way?

Equation of an ellipse Think of an ellipse as a unit circle stretched by factor a in the x-direction and factor b in the y-direction This means the equation can be given by… or Where would it cut the x and y axis?

Conic Sections (3) Parabola A parabola is formed when i.e. when the plane  is parallel to a generator.

Parabola The general equation of a parabola is given by:

Conic Sections (4) Hyperbola A hyperbola is formed when i.e. when the plane  cuts both the cones, but does not pass through the common vertex.

Hyperbola (rectangular) What type of graph is the one below? Can you write down a general equation for this type of graph? It is a reciprocal graph The point closest to the origin has co-ordinates (c,c). What are the asymptotes of this graph?

Hyperbola (rectangular) It can be shown using a matrix transformation that after a rotation of 450, our hyperbola has the general equation.. What are the asymptotes of this graph? In a rectangular hyperbola, the asymptotes always cross each other at right angles.

Hyperbola (non-rectangular) The following equation represents a hyperbola with centre at the origin and cutting the x-axis at a and –a. The asymptotes have equations: