Geometry of Conic Sections

Task goals

• Identify a conic (parabola, circle, ellipse, or hyperbola) from its standard-form equation.
• Sketch a parabola, ellipse, or hyperbola from its standard-form equation, marking the vertex or centre, the axes, and the intercepts.
• State the coordinates of the focus (or foci) and the equation of the directrix (or directrices) of a conic given in standard form.
• Derive the standard-form equation of a parabola from its focus-directrix geometric definition.
• Derive the standard-form equation of an ellipse or hyperbola from the sum-of-focal-distances or difference-of-focal-distances geometric definition.
• Use completing the square to convert a non-standard conic equation into standard form and extract its centre, vertices, and axis lengths.
• Calculate the eccentricity of an ellipse or hyperbola and interpret what its value indicates about the conic's shape.
• Solve a problem involving the intersection of a conic and a line, such as finding a tangent or the points of intersection, justifying the method algebraically.
• Apply conic geometry to a real-world context, such as a satellite dish, planetary orbit, or whispering gallery, to find an unknown length, position, or dimension.