🛰️ AINMATH · Classroom Worksheet
The Parabola
Middle–high school · Conic Sections
⏱ 15–20 min
Class
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Learning goal — Understand what the focus and directrix of a parabola mean, be able to find the focus and directrix from an equation, and confirm that a point on the parabola is equidistant from the focus and the directrix.
Parabola x² = 4py → Focus (0, p) · Directrix y = −p
💻 Access on a tablet → ainmath.com/parabola.html
1Finding the Focus and Directrix
Q. Find the focus and directrix of the parabola x² = 12y.
2Confirming the Definition
Point P(4, 2) lies on the parabola x² = 8y.
① Find this parabola's focus and directrix.
② Find the distance from point P to the focus and the distance from P to the directrix, and confirm the two values are equal.
3Think Further — Real-World Connections
Q. Using the reflective property of a parabola, explain why satellite dishes are shaped like a paraboloid (a parabola rotated around its axis).
For teachers — cover before handing out to students
Answers & Solutions
Activity 1 — 4p=12 → p=3. Focus (0, 3), directrix y=−3
Activity 2 — ① 4p=8 → p=2. Focus (0, 2), directrix y=−2 ② Distance to focus = √((4−0)²+(2−2)²) = 4, distance to directrix = 2−(−2) = 4. Confirms the two values are equal.
Activity 3 — Radio waves from a satellite come from so far away that they can be treated as parallel rays. Light (or radio waves) traveling parallel to a parabola's axis, once reflected off the paraboloid surface, all converge at the single focus point, which lets a weak signal be concentrated at one point (the receiver) to make it strong.