Conic Sections - Examples & Applications

This section provides step-by-step examples on how to work with the equations of conic sections. You will learn to identify the type of conic from a general equation, convert equations into standard form by completing the square, and extract key geometric properties like the center, vertices, and foci.

The general equation $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ can be converted to standard form by completing the square.

The general equation for a conic section is $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$. Assuming the axes are not rotated ($B = 0$), you can quickly identify the shape by looking at the coefficients $A$ and $C$.

The standard form for a circle is $(x - h)^2 + (y - k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

The standard form for a vertical parabola is $(x - h)^2 = 4p(y - k)$. For a horizontal parabola, it is $(y - k)^2 = 4p(x - h)$.

The standard form of an ellipse is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. The larger denominator determines the major axis.

Each type of conic section has unique properties—foci, directrices, asymptotes—that can be derived from its standard form equation.