Ellipse

Definition. An ellipseis the locus of points for which the sum of distances to two fixed points is constant and equal to 2 а.

Take two fixed points at a distance 2c apart, join them by a straight line, and extend this line to the x -axis. We draw the perpendicular line through the center of the segment between the focuses and take it for one coordinate axis.

Let us derive the equation of the ellipse.

у

M₄( 0; b)

r₁ М(х;у)

r₂

M₁(–a; 0 ) F₁ (–c; 0 ) 0 F₂ (c; 0 ) M₂(a; 0 ) х

M₃( 0; –b)

 

 

The points F ₁ and F ₂ are called the foci of the ellipse, and r ₁ and r ₂ are its focal radii.

To derive the equation of an ellipse, we take an arbitrary point М(х,у) and consider the distances to the foci:

, .

 

The characteristic feature of this line is, by definition,

.

This is the equation of the ellipse. Let us reduce it to a convenient form:

,

.

Eliminating some terms and reducing by 4, we obtain

.

Let us square both sides:

We obtain

.

Let us divide both sides by :

;

changing the sign, we obtain the equation of the ellipse:

Since the length 2 a of a polygonal line is larger than the length 2 c of a straight line, we can denote the difference of squares by

. (*)

Thus, we obtain the classical equation of an ellipse:

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