EQUATION OF THE LOCUS
Equation of the locus: A locus is a set of points that satisfy certain geometric conditions. Many geometric shapes are most naturally and easily described as loci. For example, a circle is the set of points in a plane that are a fixed distance from a given point to the center of the circle.
Locus Of Points
A locus of points is the set of points, and only those points, that satisfies given conditions. The locus of points at a given distance from a given point is a circle whose center is the given point and whose radius is the given distance. In analytic geometry, a curve on a graph is the locus of analytic points that satisfies the equation of the curve. For example, the locus of points such that the sum of the squares of the coordinates is a constant is a circle whose center is the origin. The constant is the square of the radius, and the equation of the locus (the circle) is.
Equation of the locus intermediate mathematics 1B
Problems involving describing a certain locus can often be solved by explicitly finding equations for the coordinates of the points in the locus. Here is a step-by-step procedure for finding plane loci:
Step 1: If possible, choose a coordinate system that will make computations and equations as simple as possible.
Step 2: Write the given conditions in mathematical form involving the coordinates .
Step 3: Simplify the resulting equations.
Step 4: Identify the shape cut out by the equations.
Five fundamental locus theorems and how to use them
Locus Theorem 1: The locus of points at a fixed distance, d, from the point, P is a circle with the given point P as its center and d as its radius.
Locus Theorem 2: The locus of the points at a fixed distance, d, from a line, l, is a pair of parallel lines d distance from l and on either side of l.
Locus Theorem 3: The locus of points equidistant from two points, P and Q, is the perpendicular bisector of the line segment determined by the two points.
Locus Theorem 4: The locus of points equidistant from two parallel lines, l1 and l2, is a line parallel to both l1 and l2 and midway between them.
Locus Theorem 5: The locus of points equidistant from two intersecting lines, l1and l2, is a pair of bisectors that bisect the angles formed by l1 and l2.
Example 1: A treasure map shows a treasure hidden in a park near a tree and a statue. The map indicates that the tree and the stature are 10 feet apart. The treasure is buried 7 feet from the base of the tree and also 5 feet from the base of the stature. How many places are possible locations for the treasure to be buried? Draw a diagram of the treasure map, and indicate with an X each possible location of the treasure.
Example 2: The distance between the parallel line l and m is 12 units. Point A is on line l. How many points are equidistant from lines l and m and 8 units from point A.
Example 3: Maria's backyard has two trees that are 40 feet apart. She wants to place lampposts so that the the posts are 30 feet from both of the trees. Draw a sketch to show where the lampposts could be placed in relation to the trees. How many locations for the lampposts are possible
Five rules of locus theorem using real world examples
Locus is a set of points that satisfy a given condition.
There are five fundamental locus rules.
Rule 1: Given a point, the locus of points is a circle.
Rule 2: Given two points, the locus of points is a straight line midway between the two points.
Rule 3: Given a straight line, the locus of points is two parallel lines.
Rule 4: Given two parallel lines, the locus of points is a line midway between the two parallel lines.
Rule 5: Given two intersecting lines, the locus of points is a pair of lines that cut the intersecting lines in half.