3. Second-Order Curves in the Plane Ellipse
Definition 3. An ellipse is the locus of points for which the sum of distances to two fixed points is constant and equal to 2а.
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 F1 and F2 are called the foci of the ellipse, and r1 and r2 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 2a of a polygonal line is larger than the length 2c of a straight line, we can denote the difference of squares by
.
Thus, we obtain the classical equation of an ellipse:
.
Hyperbola
Definition 4. The locus of the points for which the difference of distances to two fixed points is constant equal to 2а is called a hyperbola.
As for an ellipse, we introduce a new coordinate system:
у
М(х,у)
r1
r2
F1(–c,0) M1 M2 F2(c,0) x
To derive the equation of a hyperbola, we take an arbitrary point М(х,у) on the hyperbola and consider the distances from this point to the foci:
;
.
The characteristic feature of the line is, by definition,
.
We have composed an equation of the hyperbola. Let us reduce to a convenient form (by analogy with the ellipse):
.
We divide both sides by :
Changing the sign, we obtain the equation of a hyperbola:
.
Since 2a<2c, we denote the difference of squares by
.
Thus, we have obtained the classical equation of a hyperbola:
.