# High School: Geometry

### Expressing Geometric Properties with Equations HSG-GPE.A.2

2. Derive the equation of a parabola given a focus and directrix.

What do your satellite dish, the path that Katniss Everdeen's arrow would follow if she aimed up in the sky, and the water swirling around inside a flushing toilet all have in common? Parabolas. Yes, you heard us right. The shape you get from slicing a cone perpendicularly to its base shows up in plenty of unusual real-world places.

Students should already know that parabolas, like pretty much everything else in the Mathland, can be labeled with their own unique equations in the form y = ax2 + bx + c. But giving your students the values of a, b, and c would just be too easy, don't you think? Instead, have them find the equation when given a focus and directrix.

Students should know that a focus is a point near a parabola and a directrix is a line near a parabola. So what's so special about them? In Mathese, the parabola is the set of all the points that are the same distance between two things: a given point and a given line. In picture form, that would be something like this.

Students should know that because the distances are equal, they find the equation of a parabola by applying the bulky equation below, where xf and yf are the coordinates of the focus and either xd or yd can be substituted with the value of the directrix. The other value becomes its respective variable (so either xd becomes x, or yd becomes y).

To help students become more efficient, tell them to square both sides and then expand them. Then, they can solve for y to get the equation of the parabola. No sweat (unless they happen to be on a treadmill or something).

Students can also use the (x – h)2 = 4p(yk) formula for a horizontal directrix and (y – k)2 = 4p(xh) for a vertical directrix. They should know that (h, k) are the coordinates of the parabola's vertex and p is the distance from the focus to the vertex. Remind students that p will be negative when the focus is either under or to the left of the directrix.

Sideways parabolas may exist in the real world when the satellite dish gets knocked over or when you stand on a platform and throw a boomerang. Hopefully, the watery flushing toilet parabola never goes sideways.

That would be gross.

#### Drills

1. What is the equation of the parabola with focus (5, 7) and directrix y = 2?

y = 0.1x2x + 7

Plugging the coordinates of the focus (5, 7) and the general coordinates of the directrix (x, 2) in to the equal distances equation and then simplifying yields the equation y = 0.1x2x + 7. Answer choice (A) reflects using the negatives of those coordinates, while choices (C) and (D) result from improper combination of terms.

2. What is the equation of the parabola with focus (-4, 9) and directrix y = 6?

If we replace the x's and y's in the equal distances equation with the coordinates of the focus (-4, 9) and directrix (x, 6), we get the equation shown in (C). Choice (A) uses 4 instead of -4 while (B) uses 4 instead of -4 and -9 instead of 9. Answer (D) incorrectly used x = 6 as the directrix.

3. What is the equation of the parabola with focus and directrix ?

Simplifying the equal distances equation after plugging in the focus and directrix gives us the equation . In answer choice (A), the negative of the focus coordinates were used. Choice (B) reflects an incorrect combining of terms, and choice (C) is really just a random guess.

4. Obviously distracted by Peeta's witty charm, Katniss's shot missed by a mile when her arrow followed the trajectory of the parabola with focus (75, 125) and directrix y = 150. Which equation describes the arrow's trajectory?

If we plug the focus and directrix coordinates in to the equal distances equation and simplify, we should get (A). Just looking at the equations, we know the coefficient of the x2 term has to be negative, so (B) and (D) aren't right. Choice (C) combines the terms incorrectly.

5. A boomerang follows the trajectory of the parabola with the focus and directrix Which equation matches that parabola?

Since the directrix is a vertical line, not a horizontal one, we know that our parabola should take the form x = ay2 + by + c. That narrows our options down to (C) or (D). Using the coordinates of the focus and the general coordinates of the directrix, the equal distances equation simplifies to the equation given in (C). We may have ended up with (A) or (B) if we switched the x and y. Option (D) also has issues with the pluses and minuses.

6. Whoever installed our IndirecTV satellite dish clearly wasn't thinking, since it is quite obviously aimed the wrong direction. It's aligned with the parabola with the focus (3, 2) and directrix y = 2x for when x > 1. Which equation describes this curve?

In this case, we solve as we normally would but instead of using a number, we use 2x for the directrix. Simplifying should give us (B). Choice (A) reflects using the negatives of the focus coordinates and we'd get (C) if we carelessly combined terms. Even though technically, only (D) describes a true parabola, it's not applicable to this problem.

7. Which equation matches this parabola?

y = 0.1x2 – 2.4x + 26.9

The focus is at (12, 15) and the directrix is y = 10. We can also use our general knowledge of parabolas to realize that the coefficient of x2 should be positive and that its y-intercept is at 26.9 rather than 269. Or, you know, we could use the focus and directrix. Answer (C) has errors with the pluses and minuses, while (A) or (D) would still need to be divided by 10.

8. Which equation matches this parabola? (The directrix is line y = -2.)

We can clearly see that our focus is at (-4, -5) and the directrix is at y = -2. Those values should give us the equation in (D). Choice (B) reflects entering the negative coordinates as positives when filling in the equal distances equation. Choices (A) and (C) reflect an error in combining like terms.

9. Which equation names the parabola that could be drawn from this focus and directrix?

This sideways parabola will have the equation given in (A) when we plug the focus and directrix into the equal distances equation. Answer (B) has the x's and y's switched. Answers (C) and (D) reflect a starting error of using a directrix of y = 4.

10. Which equation names the parabola that could be drawn from this focus and directrix?