Focus of a Parabola: Definition, Formula, Examples
Focus of a Parabola: Definition, Formula, Examples
TL;DR
The focus of a parabola is the single fixed point on the axis such that every point on the curve is the same distance from the focus as it is from the directrix line. This article covers the definition, the focus formula for all four standard forms, the focal distance, the latus rectum and focal chord, six worked examples, and the common mistakes.
What Is the Focus of a Parabola?
A parabola is the set of all points in a plane that are equidistant from a fixed point and a fixed line. The fixed point is the focus, and the fixed line is the directrix. A parabola is defined by both of these together, this article centres on the focus and its focal properties.
State the defining rule precisely. If F is the focus and P is any point on the parabola, and d is the perpendicular distance from P to the directrix, then:
PF=d for every point P on the curve.
The focus always lies inside the curve, on the axis of symmetry, the line that cuts the parabola into two mirror halves. The vertex sits exactly halfway between the focus and the directrix.
How Do You Find the Focus of a Parabola?
The focus comes straight from the standard equation. For a parabola with vertex at the origin opening to the right:
y^2=4ax ⇒ focus at (a,0), where a is the distance from the vertex to the focus, called the focal distance.
| Equation | Opens | Focus | Directrix |
|---|---|---|---|
| y^2=4a | right | (a,0) | x=−a |
| y^2=−4a | left | (−a,0) | x=a |
| x^2=4a | up | (0,a) | y=−a |
| x^2=−4a | down | (0,−a) | y=a |
The pattern is the same every time: read off 4a from the coefficient, halve it twice to get a, then step a units from the vertex along the axis, toward the inside of the curve. For a translated parabola, the vertex moves and the focus rides along with it.
Why the focus sits at distance a
Take the simplest case, vertex at the origin, focus at some unknown point (0,p) on the y-axis, directrix the line y=−p. A point (x,y) on the parabola must satisfy "distance to focus equals distance to directrix":
√(x² + (y - p)²) = y + p.
Square both sides and expand:
x² + y² - 2py + p² = y² + 2py + p².
The y² and p² terms cancel, leaving x² = 4py. Comparing with the standard form x² = 4ay shows p=a. So the focal distance a in the equation is the distance from vertex to focus.
The Focal Distance, Latus Rectum, and Focal Chord
Three focus-related lengths show up constantly, and they all trace back to that one number a.
- Focal distance of a point. For any point P on the parabola, the focal distance PF equals its distance to the directrix.
- Latus rectum. The latus rectum is the focal chord drawn perpendicular to the axis; its length is always 4a.
- Focal chord. Any chord that passes through the focus is a focal chord.
Examples of the Focus of a Parabola
Example 1 - Find the focus of the parabola y²=12x
Compare with y²=4a: here 4a=12, so a=3. Final answer: the focus is at (3,0).
Example 2 - Find the focus of the parabola x²=−8y
Final answer: the focus is at (0,-2).
Example 3 - Find the focus of y²=−20x
Final answer: the focus is at (−5,0).
Example 4 - A parabola has the equation (x−2)²=8(y+1)
Final answer: the focus is at (2,1).
Example 5 - The parabola y²=4ax has its focus at (4,0).
Final answer: the latus rectum is 16 units long.
Example 6 - A parabola opens upward with vertex at the origin and passes through (6,3).
Final answer: the focus is at (0,3).
Why the Focus of a Parabola Matters
The focus describes where technology works, influencing satellite dishes, headlights, and solar concentrators.
Where Things Go Sideways With the Focus of a Parabola
Mistake 1: Ignoring the sign and the opening direction
Mistake 2: Confusing 4a with a
Mistake 3: Forgetting to shift the focus for a translated parabola
Key Takeaways
The focus of a parabola is a fixed point where every point on the curve is equidistant from the focus and the directrix.
The latus rectum has length 4a, and for y²=4ax its endpoints are (a,2a) and (a,−2a).
The focus is why satellite dishes, headlights, and solar concentrators work.