Ellipse

Definition 3. 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.

Construction. An ellipse can be drawn as follows: we hammer nails at some distance apart, tie a fillet to the nails, and span it with a chalk. Drawing a closed line with a chalk, spanning the fillet, we obtain an ellipse (this was demonstrated by the author at his lectures).

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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