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# Tangents to an Ellipse

Consider the equation of an ellipse: у .

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х

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 М0 belongs to the ellipse, the coordinates of М0 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 , .

Hyperbola

Definition. The locus of the points for which the difference of distances to two fixed points is constant equal to 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: . (19)

Date: 2015-01-02; view: 935

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