Extensions Page 4d

Page 4d

Determining the Location of the Focal Point
and the Equation of the Directrix

How would you find the coordinates of the focus and equation of the directrix for a parabola defined by an equation of the form y = ax² + bx + c or x = ay² + by + c?
Recall that if you write an equation in the form (x - h)² = 4p(y - k) or (y - k)² = 4p(x - h), p is half the distance between the focal point and the directrix, and (h, k) represents the coordinates of the parabola's vertex. This information can then be used to calculate the coordinates of the focal point and the equation of the directrix.
Let's look at a few examples to see how this is done.

EXAMPLE 1:
The equation x² + 12y = 0 defines a parabola. What are the coordinates of the focal point and the equation of the directrix?
Write the equation in the form x² = 4py. x² = -12y
From this equation, you know that the parabola's vertex is at (0, 0) and it opens downward.

Since 12 = 4p, you can see that p = 3. Therefore, one-half the distance between the directrix and the focus is 3 units.
Since the vertex of the parabola lies half way between the focal point and the directrix, the focal point is 3 units below the vertex and the directrix is 3 units above the vertex.
Therefore, the focal point is located at (0, -3) and the directrix is the line defined by y = 3.

EXAMPLE 1:
The equation x² + 12y = 0 defines a parabola. What are the coordinates of the focal point and the equation of the directrix?
Write the equation in the form x² = 4py. x² = -12y
From this equation, you know that the parabola's vertex is at (0, 0) and it opens downward.
Since 12 = 4p, you can see that p = 3. Therefore, one-half the distance between the directrix and the focus is 3 units.
Since the vertex of the parabola lies half way between the focal point and the directrix, the focal point is 3 units below the vertex and the directrix is 3 units above the vertex.
Therefore, the focal point is located at (0, -3) and the directrix is the line defined by y = 3.

Answer the questions in the question box below to practice finding the focal point and directrix using an equation of the form ax² + bx + c = 0 or ay² + by + c = 0.