Conics

Conic sections are intersections of a plane and a double-napped cone.  These intersections do not pass through the vertex of the cones.
Circles:  planes passes through one cone parallel to the base  (x - h)2 + (y - k)2 = r2
Ellipse:  plane passes through one cone not parallel to the base
Parabolas:  plane passes through a cone and its base
Hyperbola:  plane passes through both cones

Degenerate conics pass through the vertex of the cones.
Point:  plane passes through the vertex parallel to the bases.
Line:  plane passes through the vertex down the slant sides of the cones.
Two intersecting Lines:  plane passes through the vertex perpendicular to the bases of the cones.

General form of the equation of a conic:  Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
 

Parabolas

Definition:

A parabola is a curve consisting of a set of points P whose distance from a fixed oint (focus) is equal to the perpendicular distance from P to a line (directrix) that does not contain the focus.
 

Derive the equation of a parabola:

Let the vertex be  V = (h, k)  The equation of the directrix is y = k - m, m is not equal to 0.  The focus is F = (h, k + m).  The point on the parabola is P = (x,y).
 

According to the definition,  \/((x - h)2 + (y - (k + m))2)   =   \/((x - x)2 + (y - (k - m))2)
Square both sides and simplify the radicands.
                     x2 - 2hx + h2 + y2 - 2(k + m)y + k2 + 2mk + m2 =  0 + y2 - 2(k - m)y + (k - m)2
Simplifying  ,  x2 - 2hx + h2 + y2 - 2ky - 2my + k2 + 2mk + m2 =  0 + y2 - 2ky + 2my +  k2 - 2mk + m2
Adding like terms,  x2 - 2hx + h2 - 2my + 2mk  =  0 + 2my - 2mk
Put the y terms on the left and everything on the right, - 4my =   -x2 + 2hx - h2  - 4mk
Multiply by -1,                                                             4my =   x2 - 2hx + h2 + 4mk
Subtract 4mk and factor the right side,                          4my - 4mk = (x - h)2
Divide out 4m on the left side,                                       4m(y - k) = (x - h)2  (Standard Form)
Divide by 4m                                                                      (y - k) = (1/(4m))(x - h)2
Add k                                                                                        y = (1/(4m))(x - h)2 + k

The vertex form for a parabola is y = a(x - h)2 + k,  so this indicates that a = 1/(4m) where m is the distance from the focus to the vertex or the distance from the vertex to the directix.
The general form of the equation of a parabola is y = ax2 + bx + c where the a is the same a as in the vertex form.

Things to know about 4m(y - k) = (x - h)2   or    y = ax2 + bx + c
1.  V = (h, k)
2.  Distance from the focus to the vertex = m
3.  Directix:  y = k - m when a > 0 and y = k + m when a < 0
4.  Focus:  (h, k + m) when a > 0 and (h, k - m) when a < 0
5.  Parabola opens up when a > 0 and down when a < 0

Things to know about 4m(x - h) = (y - k)2      or     x = ay2 + by + c
1.  V = (h, k)
2.  Distance from the focus to the vertex = m
3.  Directix:  x = h - m when a > 0 and x = h + m when a < 0
4.  Focus:  (h, k + m) when a > 0 and (h, k - m) when a < 0
5.  Parabola opens to the right when a > 0 and to the left when a < 0

Examples:
1.  Graph   x + 2 = -2(y - 3)2
     Vertex:  (2, 3)
     Plot a few points.

2.  Find the equation of a parabola whose vertex is (1,5) and whose focus is (1,2).
     Since the focus is directly under the vertex, this parabola opens down and the a < 0.
     The distance m from the focus to the vertex is 5 - 2 = 3, so 1/(4m) = 1/12, and the equation is
     y - 5 = (1/12)(x - 1)2

3.  Find the focus of the parabola y = 2x2 + 3x - 4.
     a = 1/(4m)
     2 = 1/(4m)
     8m = 1
       m = 1/8
     so the distance from the vertex to the focus is 1/8.
     The vertex is obtained by using x = -b/(2a) = -3/4
         and plugging this value for x into the equation produces y = 2(-3/4)2 + 3(-3/4) - 4 = 2(9/16) - 9/4 - 4
         = 9/8 - 9/4 - 4 = 9/8 - 18/8 - 32/8 = -41/8
     so the vertex is (-3/4, -41/8)
     This parabola opens up, so the focus is directly above the vertex and its coordinates would be  (-3/4, -41/8 + 1/8)
       = (-3/4, -40/8)  =  (-3/4, -5)
 

Latus Rectum:

A line segment that passes through the focus of a parabola and has endpoints on the parabola is called a focal chord.  The focal chord that is perpendicular to the parabola's axis of symmetry is the latus rectum.
 

Problems (there are no problems here yet)

Go to Precalculus Page
Go to Precalculus Lesson Page
Go to Topics in Analytic Geometry Page