# Parabola in a Cartesian Plane

## Parabola in a Cartesian Plane

In a Cartesian plane, the site of points equidistant from a fixed line called the directrix and from a fixed point called the focus.

In the Cartesian plane below, d is the directrix of the parabola, F is its focus, and S is its vertex.

### Properties

• The axis of the parabola is an axis of symmetry. This axis contains the vertex and the focus. It is called the focal axis of the parabola.

• We can say that a point P is on the parabola if and only if: d(P, F) = d(P, d),
which means if: $$\textrm{m}\overline{\textrm{PH}}$$ = $$\textrm{m}\overline{\textrm{PF}}$$.
• The vertex of a parabola is the point of intersection between the axis of the parabola and the parabola.
• All chords that pass through the focus of the parabola and that connect two points on this parabola are called focal chords. Those that are perpendicular to the focal axis are called the latus rectum of the parabola.
• All parabolas boil down, by translation or rotation, to a parabola in which the focus is on one of the axes of the Cartesian plane and in which the vertex is at the origin. The parabola of the equation $$y^{2} = 2px$$ is called the basic parabola.

(1) y$$^{2}$$ = 2px and p > 0 :

(2) y$$^{2}$$ = 2px and p < 0 :

(3) x$$^{2}$$ = 2py and p > 0 :

(4) x$$^{2}$$ = 2py et p < 0 :

If, after a translation of the coordinate system, the coordinates of the vertex are (h, k), the equation of the parabola takes one of these standard forms:

(i) The equation is (yk)$$^{2}$$ = 2p(xh), its vertex is S(h, k), is focus is F(+ $$\frac{p}{2}$$, k) and the equation of its directrix is x = h – $$\frac{p}{2}$$.

It is open to the right if p > 0 and to the left if p < 0.
The equation of its line of symmetry is y = k.

(ii) The equation is (x – h)$$^{2}$$ = 2p(y – k), its vertex is S(h, k), is focus is F(h, k + $$\frac{p}{2}$$) and the equation of its directrix is y = k – $$\frac{p}{2}$$.

It opens toward the top if p > 0 and toward the base if p < 0.
The equation of its line of symmetry is x = h.