Unit 6, Lesson 7
Distances and Parabolas
Geometry
7.1
Warm-up
What do you notice? What do you wonder?
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What do you notice? What do you wonder?
Here are several images of parabolas.
Look at the focus and directrix of each parabola. In each case, the directrix is the -axis.
The image shows a parabola with focus and directrix (the -axis).
The diagram shows several points that are the same distance from the point as they are from the line . (Distance is measured from each point to the line along a segment perpendicular to the line.) The set of all points that are the same distance from a given point and a given line form a parabola. The given point is called the parabola’s focus and the line is called its directrix.
Focus and directrix on coordinate plane, no grid. X axis from negative 3 to 7. Y axis from negative 3 to 3. Parabola opens upward with vertex at 2 comma negative 1. Points plotted at negative 2 comma 1, 0 comma negative 0 point 5, 2 comma negative 1, 4 comma negative 0 point 5, and 6 comma 1. Focus plotted and labeled at 2 comma 1. Dotted lines drawn from each point to focus. Directrix, horizontal line, y equals negative 3. Vertical lines drawn from each point to directrix.
We can use this definition to test if points are on a parabola. The image shows the parabola with focus and directrix . The point appears to be on the parabola. Counting downward, the distance between and the directrix is 5 units.
Now use the Pythagorean Theorem to find the distance between and the focus, . Imagine drawing a right triangle whose hypotenuse is the segment connecting and . The lengths of the triangle’s legs can be found by subtracting the corresponding coordinates of the points.
Use those lengths in the Pythagorean Theorem to get . Evaluate the left side of the equation to find that . The distance, then, is 5 units because 5 is the positive number that squares to make 25. Now we know the point really is on the parabola, because it’s 5 units away from both the focus and the directrix.
A directrix is the line that, together with a point called the focus, defines a parabola.
This diagram shows a parabola is the set of points equidistant from the focus and directrix.
A focus is the point that, together with a line called the directrix, defines a parabola.
This diagram shows a parabola is the set of points equidistant from the focus and directrix.
A parabola is the set of points that are equidistant from a given point, called the focus, and a given line, called the directrix.