Tangents to an Ellipse

Consider the equation of an ellipse:

у .

М₀(х₀;у₀)

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 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:

. (19)

Date: 2015-01-02; view: 1313