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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 0 belongs to the ellipse, the coordinates of 0 must satisfy its equation, and the right-hand 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

, .




Definition. The locus of the points for which the difference of distances to two fixed points is constant equal to 2 is called a hyperbola.

As for an ellipse, we introduce a new coordinate system:




r1 r2

F1(c,0) M1 M2 F2(c,0) x



To derive the equation of a hyperbola, we take an arbitrary point (,) 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: 2015-01-02; view: 1130

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