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Pre-Calculus Conic sections

Locus definitions of conic sections

20 practice questions 0 video lessons Theory + worked examples

Locus Definitions of Conic Sections

California Pre-Calculus • Standard G-GPE • Conic Sections

Locus Definitions of Conic Sections is a topic in Conic Sections in the California Common Core State Standards. It is aligned to Standard G-GPE, which requires students to use the locus definitions to derive the equations of the conic sections.

The locus definitions describe each conic by a distance condition — a fixed distance, a focus-directrix balance, or a constant sum or difference of focal distances.

California Pre-Calculus › Conic Sections › Locus Definitions of Conic Sections  —  Standard G-GPE

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Theory

A locus is the set of all points satisfying a geometric condition. Each conic has a locus definition:

  • Circle: points a fixed distance (radius) from a center.
  • Parabola: points equidistant from a focus and a directrix.
  • Ellipse: points whose distances to two foci have a constant sum.
  • Hyperbola: points whose distances to two foci have a constant difference.
These definitions generate the equations. Translating “sum of distances \(=2a\)” into algebra produces the standard ellipse equation.
Locus definitions Locus definitions Locus definitions circle: fixed distance from a point parabola: focus = directrix distance ellipse: sum of focal distances fixed hyperbola: difference of focal distances fixed
The locus definition of each conic.
Locus definition of an ellipse For any point on an ellipse, the sum of the distances to the two foci is the same constant. x y d₁+d₂ = constant
Ellipse: the sum of distances to the two foci is constant.

The defining conditions:

\[\text{ellipse: } d_1+d_2=2a,\qquad \text{hyperbola: } |d_1-d_2|=2a\]
an ellipse keeps the sum of focal distances constant; a hyperbola keeps the difference constant
Sum vs difference is the one word that separates ellipse from hyperbola.

How to use a locus definition

  1. Identify the condition (distance to point, to line, or to two foci).
  2. Match it to the conic it defines.
  3. Extract \(a\) (and \(c\)) from the constant and the focus positions.
  4. Write the standard equation using \(b^2=|a^2\mp c^2|\).
Example 1 — Locus of a circle
Describe the locus of points \(3\) units from \((1,2)\).
Solution

A fixed distance from a fixed point is a circle.

\[(x-1)^2+(y-2)^2=9\]
a circle of radius 3 centered at 1 comma 2
Example 2 — Locus of a parabola
What curve is equidistant from the point \((0,2)\) and the line \(y=-2\)?
Solution

Equal distance to a focus and directrix defines a parabola with vertex midway, at the origin, \(p=2\).

\[x^2=8y\]
a parabola x squared equals 8 y
Example 3 — Locus of an ellipse
Points whose distances to \((\pm 3,0)\) sum to \(10\) form what curve?
Solution

Sum of focal distances constant \(\Rightarrow\) ellipse, with \(2a=10\Rightarrow a=5\) and \(c=3\), so \(b^2=25-9=16\).

\[\dfrac{x^2}{25}+\dfrac{y^2}{16}=1\]
an ellipse x squared over 25 plus y squared over 16 equals 1
Example 4 — Locus of a hyperbola
Points whose distances to \((\pm 5,0)\) differ by \(6\) form what curve?
Solution

Difference of focal distances constant \(\Rightarrow\) hyperbola, with \(2a=6\Rightarrow a=3\), \(c=5\), so \(b^2=25-9=16\).

\[\dfrac{x^2}{9}-\dfrac{y^2}{16}=1\]
a hyperbola x squared over 9 minus y squared over 16 equals 1

Common pitfalls

Sum \(\to\) ellipse, difference \(\to\) hyperbola. Read the condition carefully.
The constant is \(2a\), not \(a\). Halve it to get \(a\).
Ellipse: \(b^2=a^2-c^2\); hyperbola: \(b^2=c^2-a^2\). Keep the signs straight.

Frequently asked questions

What is a locus?

The set of all points that satisfy a given geometric condition — the basis for defining each conic.

What is the locus definition of an ellipse?

All points whose distances to two fixed foci add to a constant \(2a\).

What is the locus definition of a hyperbola?

All points whose distances to two fixed foci differ by a constant \(2a\).

How does the parabola's locus definition work?

Every point is equidistant from the focus and the directrix, which forces the parabolic shape.

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