Page 100 - Calc for the Clueless

For the parabola, ellipse, and hyperbola, it is essential to relate the equation to the picture. If you do, these
curves are very simple.

Definition

Parabola—The set of all points equidistant from a point, called a focus, and a line, called a directrix. Point V is
the vertex, equidistant from the focus and directrix and closest to the directrix and to the focus.

Let's do this development algebraically. Let the vertex be at (0,0). The focus is (0,c). The directrix is y = -c. Let
(x,y) be any point on the parabola. The definition of a parabola says FP = PQ. Just like before, everything on
PQ has the same x value and everything on RQ has the same y value. The coordinates of Q are (x,-c). Since the
x values are the same, the length of PQ = y - (-c). Using the distance formula to get FP and setting it equal to
2
2
2
2
2
2
FP, we get ((x - 0) + (y -c) ) = y + c. Squaring, we get x + y - 2cy + c = y + 2cy + c . Simplifying, we
2 1/2
2
get x = 4cy.
We will make a small chart relating the vertex, focus, directrix, equation, and picture.
Vertex Focus Directrix Equation Picture Comment
2
1. (0,0) (0,c) y = -c x = 4cy The original
derivation

2
2. (0,0) (0,-c) y = c x = -4cy y replaced by
-y

2
3. (0,0) (c,0) x = -c x = 4cx x,y
interchange
in 1

2
4 (0,0) (-c,0) x = c y = -4cx x replaced
by-x in 3

If you relate the picture to the original equation, the sketching will be easy.

Example 13—