Parabola

Definition. The locus of points for which the distance to a fixed point equals the distance to a given straight line (a directrix) is called a parabola.

Let us draw the perpendicular through a fixed point to the given straight line and take it for the x -axis. From the middle point of the segment joining the focus to the given straight line we draw a perpendicular and take it for the y -axis.

p М(х;у)

r

0 x

 

To derive the equation of the parabola, we take an arbitrary point М(х;у) on it and write down the characteristic feature of a parabola as a mathematical formula.

The distance from the focus to the directrex is called the parameter of the parabola and denoted by p. Let us find the distance from the point М(х;у) to the focus:

,

and = .

By definition, these distances are equal:

.

 

Let us transform this, relation by squaring both sides:

; .

We obtain

. (20)

This is the classical equation of a parabola.

The parabola passes through the origin (0;0), because it satisfies equation (20).

Suppose that the parameter is a positive number р >0; then, since у² >0, we have x >0, and the parabola is contained in the right half-plane. If p <0, then x <0, and the parabola is contained in the left half-plane

y у

p>0 p<0

0 x 0 х

M₀(x₀,y₀)

 

 

Consider the equation of a parabola in the “school” form . Let us analyze this equation by analogy with (20): if p >0, then y> 0, and the branches of the parabola are directed upward; if p <0, then y <0, and the branches of the parabola are directed downward.

p >0 y p <0 y

0 x

 

0 x

 

The eccentricity of the parabola, that is, the ratio of the focal radius to the distance from a point to the directrix, equals 1, i.e.,

.

 

Tangent lines to a parabola. Given a point М ₀(х ₀, у ₀) 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 М ₀ (х _0; у 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 х=–5. The parabola is given by the equation , and the directrix by the equation , which means that and р=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.