Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for geometry parabola focus directrix and tangent lines? Then this Geometry lesson on Parabolas from Focus and Directrix, and Tangents explains each idea step by step, with worked examples.
Quick answer: A parabola is the set of points equally distant from a focus and a directrix, giving equations like x² = 4py with focus (0, p). Tangent lines to a circle are perpendicular to the radius at the point of contact, and the two tangent segments from an outside point are congruent.
Standards: G-GPE.A.2, G-C.A.4 · Level: Grade 10 Geometry
What is a parabola?
Take one point (the focus) and one line (the directrix) that does not contain it. The parabola collects every point whose two distances, one to the focus and one to the line, come out equal. Its vertex sits midway between the focus and the line.
How do you get the equation from the focus and directrix?
Place the focus at (0, p) and the directrix along y = −p. A point (x, y) belongs to the parabola exactly when these two distances match:
√(x² + (y − p)²) = y + p
After squaring, the y² and p² terms cancel and x² = 4py remains. A larger p gives a wider, flatter curve.
- Vertical axis: (x − h)² = 4p(y − k), with focus (h, k + p) and directrix y = k − p.
- Horizontal axis: (y − k)² = 4p(x − h), with focus (h + p, k) and directrix x = h − p.
- When p is negative the curve opens the opposite way.
How do tangent lines to a circle behave?
- The radius drawn to the touching point makes a right angle with the tangent.
- From any outside point, the two tangent segments have equal length.
- With center O and touching point T, PT² = OP² − r².
- Join the center O to the outside point P.
- Bisect OP with a perpendicular bisector to mark its midpoint M.
- With the compass at M, draw the circle that passes through both O and P.
- The new circle crosses the original at T and S; draw PT and PS.
Reason: OP is a diameter of the helper circle, so ∠OTP is an inscribed angle on a semicircle and equals 90°. That makes PT ⊥ OT.
A parabola has focus (0, 3) and directrix y = −3. Here p = 3, so x² = 12y. The point (6, 3) satisfies 36 = 12 · 3, and its distance to the focus (6) equals its distance to the directrix (6).
Using 4p as the distance to the focus. The distance from vertex to focus is p, so in x² = 12y the focus is 3 units from the vertex.
- Parabola: distance to the focus equals distance to the directrix.
- Opening up or down: (x − h)² = 4p(y − k), focus (h, k + p), directrix y = k − p.
- Opening right or left: (y − k)² = 4p(x − h), focus (h + p, k), directrix x = h − p.
- Two tangent segments from an outside point are congruent, and PT² = OP² − r².
Parabolas from Focus and Directrix, and Tangents: frequently asked questions
What is the focus of a parabola used for?
Rays parallel to the axis reflect off a parabolic dish and meet at the focus. That is why satellite dishes and headlights are shaped like parabolas.
How do I find p from an equation?
Write the equation as (x − h)² = c(y − k) or (y − k)² = c(x − h). Then 4p equals c, so p = c ÷ 4. A negative c means the parabola opens down or left.
Where is the vertex relative to the focus and directrix?
The vertex is exactly halfway between them, on the axis of the parabola. The distance from the vertex to each is p.
Why are two tangent segments from one point congruent?
The two right triangles formed with the center share the hypotenuse OP and have equal radii as legs, so by the Pythagorean Theorem or HL the tangent segments are equal.
How do I construct a tangent from a point outside a circle?
Find the midpoint of the segment joining the center and the point, draw a circle centered there through both, and connect the outside point to the two places where the circles intersect.