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 Ì_{0 }(õ_{0;}ó_{0}) 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 SecondOrder Curve and Its Classical Form
Definition. An equation of the form
(21)
is called a general equation of a secondorder curve.
Assigning particular values to the coefficients, we obtain the above equations of a hyperbola, an ellipse, and a parabola.
Date: 20150102; view: 1566
