A conic section is a curve obtained by intersecting a plane with the surface of a cone. The three main types are the parabola, ellipse and hyperbola; a circle is a special case of an ellipse.

General equation for a circle, ellipse or hyperbola

Compare the coefficient of \(x^2\), \(A\), with the coefficient of \(y^2\), \(B\).

Expanding \((x-a)^2+(y-b)^2=r^2\) gives \(x^2+y^2+dx+ey+f=0\). Completing the square reverses the conversion.

This is a circle with centre \((a,b)\) and radius \(r\).

Graphing Techniques

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Comparison of \(A\) and \(B\)	Conic classification
\(A\) and \(B\) have different signs	Hyperbola
\(A\) and \(B\) have the same sign	Ellipse
\(A=B\)	Circle (a special ellipse)

Conic Sections and Circles

Conic Sections and Circles

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