Locus
Understanding the Concept of Locus
Locus is a fundamental concept in geometry that represents a set of points that satisfy a particular condition or a set of conditions. It is essentially the path or the figure that is traced out by a point that moves according to a specific rule.
Definition of Locus
In mathematical terms, the locus is defined as:
> The locus of a point is the collection of all points that satisfy certain conditions.
These conditions can be algebraic equations or geometric properties. The locus can be a line, a curve, a circle, or any other geometric shape.
Properties of Locus
- A locus is a set of points that share a common property.
- The shape of a locus can be determined by algebraic equations.
- Every point on the locus satisfies the given condition(s), and every point that satisfies the condition(s) is on the locus.
Examples of Loci
- The set of all points that are at a fixed distance from a given point is a circle.
- The set of all points that are equidistant from two given points is the perpendicular bisector of the line segment joining the two points.
- The set of all points that are at a fixed distance from a given line is a pair of parallel lines, equidistant from the given line.
Formulas Related to Locus
When dealing with loci, we often encounter equations that represent the conditions that the points on the locus must satisfy. Here are some common formulas:
- Circle: The locus of points at a distance $r$ from a fixed point $(h, k)$ is given by the equation $(x - h)^2 + (y - k)^2 = r^2$.
- Perpendicular Bisector: The locus of points equidistant from $(x_1, y_1)$ and $(x_2, y_2)$ is the line with the equation $(x - x_1)(x_2 - x_1) + (y - y_1)(y_2 - y_1) = 0$.
- Parabola: The locus of points equidistant from a fixed point (focus) and a fixed line (directrix) is a parabola.
Table of Differences and Important Points
| Property | Circle | Perpendicular Bisector | Parabola |
|---|---|---|---|
| Definition | Set of points at a fixed distance from a center | Set of points equidistant from two points | Set of points equidistant from a point and a line |
| Equation | $(x - h)^2 + (y - k)^2 = r^2$ | $(x - x_1)(x_2 - x_1) + (y - y_1)(y_2 - y_1) = 0$ | $y^2 = 4ax$ (standard form) |
| Symmetry | About the center | About the line itself | About the axis of the parabola |
| Fixed Point | Center $(h, k)$ | None | Focus |
| Fixed Line | None | Line joining the two points | Directrix |
Examples to Explain Important Points
Example 1: Circle as a Locus
Problem: Find the locus of a point that moves so that it is always 5 units away from the point (3, 2).
Solution: The fixed point here is (3, 2), and the distance is 5 units. Using the formula for a circle, we get:
$$(x - 3)^2 + (y - 2)^2 = 5^2$$ $$(x - 3)^2 + (y - 2)^2 = 25$$
This equation represents the locus, which is a circle with a radius of 5 units centered at (3, 2).
Example 2: Perpendicular Bisector as a Locus
Problem: Find the locus of points that are equidistant from the points (1, 2) and (5, 6).
Solution: We use the formula for the perpendicular bisector. Let $(x, y)$ be any point on the locus. Then:
$$(x - 1)(5 - 1) + (y - 2)(6 - 2) = 0$$ $$(x - 1)(4) + (y - 2)(4) = 0$$ $$4x - 4 + 4y - 8 = 0$$ $$4x + 4y - 12 = 0$$ $$x + y - 3 = 0$$
This equation represents the locus, which is the perpendicular bisector of the line segment joining the points (1, 2) and (5, 6).
Example 3: Parabola as a Locus
Problem: Find the locus of points that are equidistant from the point (0, 0) and the line $y = -2$.
Solution: The fixed point (focus) is at the origin (0, 0), and the directrix is the line $y = -2$. The standard form of a parabola with the focus at the origin and the directrix parallel to the x-axis is $y^2 = 4ax$. Since the distance from the focus to the directrix is 2 units, $a = 2$. Thus, the equation of the locus is:
$$y^2 = 4(2)x$$ $$y^2 = 8x$$
This equation represents the locus, which is a parabola opening to the right with the vertex at the origin.
In conclusion, the concept of locus is a powerful tool in geometry that allows us to describe and analyze the behavior of points that move according to specific rules. Understanding the equations and properties of different loci is essential for solving geometric problems and for applications in various fields such as physics, engineering, and computer graphics.