Tangents to an Ellipse
Consider the equation of an ellipse:
ó .
Ì_{0}(õ_{0};ó_{0})
õ
As is known, the equation of a tangent to a curve is determined by the formula
.
Differentiating the equation of the ellipse as an implicit function, we obtain , whence , or .
Substituting this k, we find the equation of the tangent line:
.
Let us transform it:
,
. Dividing by , we obtain
.
Since the point Ì_{0} belongs to the ellipse, the coordinates of Ì_{0} must satisfy its equation, and the righthand side equals one.
Thus, the equation of a tangent to an ellipse is
. (18)
Example. Given the ellipse given , find the distance between its foci, eccentricity, and the equations of directrices.
Let us reduce the equation it to the classical form (17):
; ;
, .
Let us find the eccentricity:
The equations of the directrices are
, .
Hyperbola
Definition. The locus of the points for which the difference of distances to two fixed points is constant equal to 2à is called a hyperbola.
As for an ellipse, we introduce a new coordinate system:
ó
Ì(õ,ó)
r_{1 }r_{2}
F_{1}(–c,0) M_{1} M_{2} F_{2}(c,0) x
To derive the equation of a hyperbola, we take an arbitrary point Ì(õ,ó) 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:
. (19)
Date: 20150102; view: 1272
