# Analyzing Polar Equations for Conic Sections

## Key Questions

• There are two basic kinds of parabola(as it is convenient for me to say)

Type 1:
The parabola lying on, or parallel to the $x -$axis

This parabola is of the form ${\left(y - {y}_{v}\right)}^{2} = 4 a \left(x - {x}_{v}\right)$

Where,
- focus is $\left(a + {x}_{v} , {y}_{v}\right)$
- directrix is the line $x = {x}_{v} - a$
- Vertex is $\left({x}_{v} , {y}_{v}\right)$

Type 2:
The parabola lying on, or parallel to the $y -$axis

This parabola is of the form ${\left(x - {x}_{v}\right)}^{2} = 4 a \left(y - {y}_{v}\right)$

Where,
- focus is $\left({x}_{v} , a + {y}_{v}\right)$
- directrix is the line $y = {y}_{v} - a$
- Vertex is $\left({x}_{v} , {y}_{v}\right)$

The directrix is the vertical line $x = \frac{{a}^{2}}{c}$.

#### Explanation:

For a hyperbola ${\left(x - h\right)}^{2} / {a}^{2} - {\left(y - k\right)}^{2} / {b}^{2} = 1$,

where ${a}^{2} + {b}^{2} = {c}^{2}$,

the directrix is the line $x = {a}^{2} / c$.