Intersection points of an ellipse with the coordinates axes. To find the intersection points of an ellipse with the x-axis, we must solve the system of equations

We obtain two vertices of the ellipse: М ₁(–а; 0), М ₂(а; 0),

is called the major axis of the ellipse;

а is the major semi axis

We find the intersection of the ellipse with the y -axis by solving the system

We obtain the two other vertices of the ellipse, М ₃(0;– b) and М ₄(0; b).

is called the minor axis of the ellipse, and

b is the minor semi axis.

It is seen from equation (17) and the figure that the ellipse is symmetric with respect to the axes Ox and .

The eccentricity and directrix of an ellipse. Consider the focal radii of an ellipse

; .

By definition, we have

.

Consider the difference of squares

;

,

or .

y

d₁ М(х;у) d₂

r₁ r₂

F₁(–c; 0 ) 0 F₂(c; 0 ) x

 

x =– l x = l

 

to determine the focal radii, we solve the system of equations

or

Definition. The ratio of distances between the foci to the sum of focal radii is called eccentricity:

.

If the distance between the foci is less than 2 а, then the eccentricity is
.

Thus, the focal radii of the ellipse are

,

.

Definition. The directrix of an ellipse is the straight line parallel to the y -axis such that the ratio of the focal radius to the distance from an ellipse point to it is constant and equal the eccentricity.

Let us draw two straight lines x=–l and x=l parallel to the y -axis and find l such that the ratio of the focal radius to the distance from a point М to this straight line is constant and equals the eccentricity:

.

Substituting the distance and the focal radius, we obtain

.

The ratio is equal to the eccentricity when , i.e., is the directrix. By analogy, we obtain equations of the directrices:

; ,

where .