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Analytic geometry - Conic sections
 Construction of the tangent at the point on the parabola and from a point exterior to the parabola
 Construction of the tangent at the point on the parabola

  As the vertex of the parabola is the midpoint of the line segment whose endpoints are, the projection P1

of the given point to the axis of the parabola and the point opposite it, draw the tangent through points B and P1.

Construction of the tangents from a point exterior to the parabola

Draw the circle centered at the point A outside the parabola through the focus.

The circle intersects the directrix at points,
D1  and  D2.

Tangents from the point A are then perpendicular 
bisectors of the line segments,
D1F and D2F.

Thus, the tangency points, P1 and P2 are equidistant from the focus and the directrix.
Parabola and line examples

Example:  Find the point on the parabola  y2 = 9x closest to the line  9x + 4y + 24 = 0.

Solution:  The tangency point of the tangent parallel to the given line is the closest point.

9x + 4y + 24 = 0,   y = -(9/4)x - 6mt = - 9/4

The slope of the tangent must satisfy tangency condition of the parabola

p = 2mc  <=   mt = - 9/4,   p = 9/2
9/2 = 2 (-9/4) c   =>    c = - 1
therefore, the tangent   t ::    y = -(9/4)x - 1.

The solution to the system of equations of the tangent and the parabola gives the tangency point, that is

Example:  Given is the polar  4x + y + 12 = 0 of the parabola  y2 = -4x, find coordinates of the pole and write equations of the corresponding tangents.

Solution:  Intersections of the polar and the parabola are the tangency points of tangents drawn from the pole P.  Thus, by solving the system of equations of the polar and the parabola we get tangency points.

(1) 4x + y + 12 = 0,  =>  (2)  (-4x - 12)2 = -4x
(2)  y2 = -4x                     4x2 + 25x + 36 = 0,

Equation of the tangent at the point on the parabola,

The intersection of tangents is the pole P. Therefore, we solve the system formed by their equations,

 Conics, a family of similarly shaped curves - properties of conics

By intersecting either of the two right circular conical surfaces (nappes) with the plane perpendicular to the axis of the cone the resulting intersection is a circle c, as is shown in the right figure.

When the cutting plane is inclined to the axis of the cone at a greater angle than that made by the generating segment or generator (the slanting edge of the cone), i.e., when the plane cuts all generators of a single cone, the resulting curve is the ellipse e.

Thus, the circle is a special case of the ellipse in which the plane is perpendicular to the axis of the cone.
If the cutting plane is parallel to any generator of one of the cones, then the intersection curve is the parabola p.
When the cutting plane is inclined to the axis at a smaller angle than the generator of the cone, i.e., if the intersecting plane cuts both cones the hyperbola h is generated.

 ::  Dandelin spheres proof of conic sections focal properties

Proof that conic section curve is the ellipse

In the case when the plane E intersects all generators of the cone, as in down figure, it is possible to inscribe two spheres which will touch the conical surface and the plane.

Upper sphere touches the cone surface in a circle k1 and the plane at a point F1. Lower sphere touches the cone surface in a circle k2 and the plane at a point F2. Arbitrary chosen generating line g intersects

the circle k1 at a point M, the circle k2 at a point N and the intersection curve e at a point P.
The points, M and F1 are the tangency points of the upper sphere and points, N and F2 are the tangency points of the lower sphere of the tangents drawn from the point P exterior to the spheres.
Since the segments of tangents from a point exterior to sphere to the points of contact, are equal
PM = PF and  PN = PF2.

And since planes of circles k1and k2, are parallel, then are all corresponding generating segments equal

MN = PM + PN   is constant.

Thus, the intersection curve is the locus of points  in the plane for which sum of distances from the two fixed points F1and F2, is constant, i.e., the curve is the ellipse.

The proof due to the French/Belgian mathematician Germinal Dandelin (1794 1847).
Conics, a family of similarly shaped curves

A conic is the set of points P in a plane whose distances from a fixed point F (the focus) and a fixed line (the directrix), are in a constant ratio. This ratio named the eccentricity e determines the shape of the curve.

We can see that conics represent a family of similarly shaped curves if we write their equations in vertex form. Recall the method we used to transform equations of the ellipse and the hyperbola from standard to vertex form. We placed the vertex of the curve at the origin translating its graph.
The equation of conics in the vertex form

Thus, obtained are their vertex equations

y2 = 2px - (p/a)x2
- the ellipse and the circle
(for the circle  p = a = r)

y2 = 2px
- the parabola

y2 = 2px + (p/a)x2
- the hyperbola

In the above vertex equations we can make following substitutions for,

 the ellipse
 the circle   p = a = b = r   =>    e = 0
 the parabola e = 1
the hyperbola
Thus, the equation of conics in vertex form is   y2 = 2px - (1 - e2)x2.
 
 
 
 
 
 
 
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