Writing Polar Equations for Conic Sections

Key Questions

• General form of the conic equation

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

The coefficients $A$ and $C$ are need to identify the conic sections without having to complete the square.

$A$ and $C$ cannot be $0$ when making this determination.

Parabola$\to A \cdot C = 0$

Circle$\to A = C$

Ellipse$\to A \cdot C > 0 \mathmr{and} A \ne C$

Hyperbola$\to A \cdot C < 0$

• Standard form equations for the hyperbola.

${x}^{2} / {a}^{2} - {y}^{2} / {b}^{2} = 1$

The foci are located on the $x$-axis also called the transverse .

${y}^{2} / {a}^{2} - {x}^{2} / {b}^{2} = 1$

The foci are located on the $y$-axis also called the transverse .

• Standard form for the equation of a parabola is the same as standard for for a quadratic function:
$y = a {x}^{2} + b x + c$

Or
$f \left(x\right) = a {x}^{2} + b x + c$.

For graphing, many prefer the vertex form

$y = a {\left(x - h\right)}^{2} + k$

A conic section is a section (or slice) through a cone.

Explanation:

Depending on the angle of the slice, you can create different conic sections,

(from en.wikipedia.org)

If the slice is parallel to the base of the cone, you get a circle.

If the slice is at an angle to the base of the cone, you get an ellipse.

If the slice is parallel to the side of the cone, you get a parabola.

If the slice intersects both halves of the cone, you get a hyperbola.

There are equations for each of these conic sections, but we will not include them here.