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Tangent lines to a parabola. Given a point Ì0(õ0,ó0) on a parabola, it is required to write the equation of a tangent to the parabola at this point.

Let us find the slope of the tangent:

 

,

To this end, we differentiate equation (20) as an implicit function:

, whence , or .

Substituting this into the equation of a straight line with given slope, we obtain

; .

Since the point Ì0 (õ0;ó0) belongs to the parabola, its coordinates satisfy the equation of the parabola:

 

; or .

Thus, we obtain the equation of a tangent to the parabola

.

Example. Write the classical equation of the parabola with directrix õ=–5 . The parabola is given by the equation , and the directrix by the equation , which means that and ð=10.

Then, the required equation of a parabola is .

 

Definition. The locus of points for which the ratio of the distances to focal radii to the distances to the corresponding directrices is constant and equal to the eccentricity

, which is

(1) less than 1, then it is called an ellipse;

(2) larger than 1, is called a hyperbola;

(3) equal to 1, is called a parabola.

 

A General Equation of a Second-Order Curve and Its Classical Form

 

Definition. An equation of the form

(21)

is called a general equation of a second-order curve.

Assigning particular values to the coefficients, we obtain the above equations of a hyperbola, an ellipse, and a parabola.


Date: 2015-01-02; view: 1490


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