Center axis

revolving line

Name ________________________________ Introduction to Conic Sections This introduction to conic sections is going to focus on what they are and some of the skills needed to work with their equations and graphs. This year, we will only work with circles and parabolas. Physical Definitions A double cone is formed when a line is revolved around an axis. All conic sections come from the shape exposed when you cut off a section of a cone. How you cut the cone determines the shape. Parabola: The slice is made by cutting parallel to the revolving line. Circle: The slice is made by cutting perpendicular to the center axis. Ellipse: The slice is made by cutting more shallowly than the revolving line (won’t go through the

bottom). Hyperbola: The slice is made by cutting more steeply than the revolving line (will go through the

bottom).

Conic sections

are also called

quadratic relations. The standard form of a quadratic relation is Ax2 +Bxy+Cy2 +Dx +Ey = F For most of our work, coefficient B will be zero. This is, in general, the mechanism that rotates the conic sections and will be studied in more depth in a later course. It should be noted that most conic sections are not functions, only relations.

Name_____________________________________ Honors Math 2 Geometric (Locus) Definition of a Parabola A parabola is a locus defined in terms of a fixed point, called the focus, and a fixed line, called the directrix. A parabola is the set of all points P(x, y) whose distance from the focus (F) equals its distance from the directrix. In other words, PF = PD (where D is the point on the directrix closest to P – so PD is perpendicular to the directrix). The figure shows a parabola with its focus at (0, 1) and a directrix of

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y = −1. Equally spaced concentric circles with their center at the parabola’s focus enable you to measure distances from the focus. Equally spaced horizontal lines parallel to the parabola’s directrix enable you to measure vertical distances from the directrix. This type of graph paper is called focus-directrix graph paper. Notice that P is the point of intersection of the circle centered at (0, 1) with a radius of 6 and the horizontal line 6 units above the directrix. Thus, P is equidistant from the focus and directrix. Examine the figure and note that all points on the parabola are equidistant from the focus and the directrix.

Example:

In the diagram, a focus has been drawn at the point (0, 3) and a directrix has been drawn, y = –3.

a. Plot the points that are equidistant from the focus and directrix using the concentric circles and the grid. b. Identify the vertex of the resulting parabola. c. Identify p, the distance from the focus to the vertex (and the distance from the directrix to the vertex).

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66

D

P

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Deriving the Algebraic Definition of a Parabola For now, let’s look at parabolas whose focus is the point (0, p) and whose directrix is y = – p. This will mean the vertex of the parabola is at the origin, as shown in the sketch below.

If PF = PD, derive the standard equation for any point P on this parabola using the distance formula.

What if the directrix is above the focus? Draw a picture similar to the one above. Derive the standard equation for this case. What if the directrix is a vertical line rather than a horizontal one? Consider both cases.

y = -p D (x, -p)

P (x, y)

F (0, p)

Example: Graph

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x = −18y 2 and identify the vertex, focus and directrix.

Example: Write the standard equation of a parabola with its vertex at the origin and with the directrix y = 4. If you need to graph a horizontal parabola on your calculator, solve for y and entered both equations into your calculator. Example: Solve for y, put the following relation in your calculator, and sketch on the grid.

y2 + 4x = 0

HOMEWORK For the following three problems:

a. Plot the points that are equidistant from the focus and directrix using the concentric circles and the grid. b. Identify the vertex of the resulting parabola. c. Identify p, the distance from the focus to the vertex (and the distance from the directrix to the vertex). d. Write the equation of the parabola.

1. Focus at (0, –3)

Directrix of y = 3 2. Focus at (3, 0)

Directrix of x = –3

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3. Focus at (–3, 0)

Directrix of x = 3 For the next three problems: Write the standard equation for the parabola with the given characteristics. 4. Given its focus at (1,0) and 5. Given its vertex at (0, 0) 6. Given its vertex at (0, 0) its directrix at 𝑥 = −1 and its focus at (0, -5) and its directrix x = 2

7. Graph

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x =120

y 2 and identify the vertex, focus and directrix.

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