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Focus (conic section)

Focus (conic section)

A special point used to construct and define a conic section. A parabola has one focus. An ellipse has two, and so does a hyperbola. A circle can be thought of as having one focus at its center.

 

 

See also

Focus of a parabola, foci of an ellipse, foci of a hyperbola

Key Formula

d(P,F)d(P,ℓ)=e\frac{d(P, F)}{d(P, \ell)} = e
Where:
  • PP = Any point on the conic section
  • FF = The focus of the conic section
  • ℓ\ell = The directrix (a fixed line)
  • ee = The eccentricity: e < 1 for an ellipse, e = 1 for a parabola, e > 1 for a hyperbola
  • d(P,F)d(P, F) = Distance from point P to the focus F
  • d(P,ℓ)d(P, \ell) = Perpendicular distance from point P to the directrix

Worked Example

Problem: A parabola has the equation y² = 12x. Find its focus.
Step 1: Recognize the standard form. A parabola opening to the right has the form y² = 4px, where p is the distance from the vertex to the focus.
y2=4pxy^2 = 4px
Step 2: Match coefficients by comparing y² = 12x with y² = 4px.
4p=12  ⟹  p=34p = 12 \implies p = 3
Step 3: Since the vertex is at the origin (0, 0) and the parabola opens to the right, the focus is located p units to the right of the vertex along the x-axis.
F=(p, 0)=(3, 0)F = (p,\, 0) = (3,\, 0)
Answer: The focus of the parabola y² = 12x is at the point (3, 0).

Another Example

Problem: An ellipse has the equation x²/25 + y²/9 = 1. Find its two foci.
Step 1: Identify a² and b² from the standard form x²/a² + y²/b² = 1. Here a² = 25 and b² = 9, so a = 5 and b = 3.
a2=25,b2=9a^2 = 25,\quad b^2 = 9
Step 2: For an ellipse, the relationship between a, b, and c (the distance from center to each focus) is c² = a² − b².
c2=25−9=16  ⟹  c=4c^2 = 25 - 9 = 16 \implies c = 4
Step 3: Since a² is under x², the major axis is horizontal. The foci lie along the x-axis, each c units from the center (0, 0).
F1=(−4, 0),F2=(4, 0)F_1 = (-4,\, 0),\quad F_2 = (4,\, 0)
Answer: The two foci of the ellipse are at (−4, 0) and (4, 0).

Frequently Asked Questions

How many foci does each conic section have?
A parabola has exactly one focus. An ellipse and a hyperbola each have two foci. A circle is a special case of an ellipse where the two foci coincide at the center, so it can be thought of as having one focus.
What is the difference between a focus and a directrix?
A focus is a fixed point, while a directrix is a fixed line. Together, they define a conic section: every point on the curve has a constant ratio between its distance to the focus and its perpendicular distance to the directrix. This ratio is the eccentricity.

Focus vs. Center

The focus and center of a conic are different points (except for circles). For an ellipse with equation x²/a² + y²/b² = 1, the center is at the origin, but the two foci are at (±c, 0) where c² = a² − b². The center is the midpoint between the foci. For a parabola, there is no traditional center at all — only a vertex and a focus.

Why It Matters

The focus has a direct physical meaning: a parabolic mirror reflects all incoming parallel rays to its focus, which is why satellite dishes and flashlights use parabolic shapes. Planets orbit the Sun along ellipses with the Sun at one focus, a fact described by Kepler's first law. Understanding foci also helps you write equations of conics and solve problems in coordinate geometry.

Common Mistakes

Mistake: Using c² = a² − b² for a hyperbola instead of c² = a² + b².
Correction: The focus formula differs by conic type. For an ellipse, c² = a² − b². For a hyperbola, c² = a² + b². Mixing these up gives the wrong focus location.
Mistake: Assuming the focus is always on the x-axis.
Correction: The foci lie along the major axis for an ellipse and the transverse axis for a hyperbola. If the larger denominator is under y², the foci are on the y-axis, not the x-axis. Always check which variable has the larger denominator.

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