Tangents to an Ellipse
Consider the equation of an ellipse:
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 Ì0 belongs to the ellipse, the coordinates of Ì0 must satisfy its equation, and the right-hand side equals one.
Thus, the equation of a tangent to an ellipse is
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
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:
F1(–c,0) M1 M2 F2(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:
Date: 2015-01-02; view: 858