LOCUS
Drawing a Locus Satisfying One Condition
A locus in a plane is the set of all points in a plane that satisfy a given condition or a set of given conditions. The word locus is derived from the Latin word for “location”. The plural of locus is Loci, pronounced “low-sigh”.
A locus is often described as the path of an object moving in a plane. For instance, the reason that many clock faces are circular is that the locus of the end of a clock’s minute hand is a circle.
Finding a Locus
To find the locus of points that satisfy a given condition, use the following steps.
1. Draw any figures that are given in the statement of the problem.
2. Locate several points that satisfy the given condition.
3. Condition drawing points until you can recognize the pattern.
4. Draw the locus and describe it in words.
Example :
Draw point C on a piece of paper. Draw and describe the locus of all points on the paper that are e inches from c.
Solution :
Step 1 :
Draw point C. Locate several points e inches
from C.
Step 2 :
Recognize a pattern: the points lie on a circle.
Step 3 :
Draw the circle using compass as shown below.
The locus of points on the paper that are e inches from C is a circle with center C and a radius of 3 inches.
Loci Satisfying Two or More Conditions
To find the locus of points that satisfy two or more conditions, first find the locus of points that satisfy each condition alone. Then find the intersection of these loci.
Example 1 :
Points A and B lie in a plane. What is the locus of points in the plane that are equidistant from points A and B and are a distance of AB from B ?
Solution :
The locus of all points that are equidistant from A and B is the perpendicular bisector of AB as shown below.
The locus of all points that are a distance of AB from B is the circle with center B and radius AB as shown below.
These loci intersect at D and E. So D and E are the locus of points that satisfy both conditions.
Example 2 :
Point P is in the interior of ∠ABC. What is the locus of points in the interior of ∠ABC that are equidistant from both sides of ∠ABC and 2 inches form P ? How does the location of P within ∠ABC affect the locus ?
Solution :
The locus of points equidistant from both sides of ∠ABC is the angle bisector. The locus of points 2 inches form P is a circle. The intersection of the angle bisector and the circle depends on the location of P. The locus can be 2 points, 1 point, or 0 points.
The locus is two points :
The locus is one point :
The locus is zero points :
Finding a Locus Satisfying Three Conditions
Example :
We have the following readings of an earthquake from three seismographs.
(i) At A(-5, 5), the epicenter is 4 miles away.
(ii) At B(-4, -3.5), the epicenter is 5 miles away.
(iii) At C(1, 1.5), the epicenter is 7 miles away.
Where is the epicenter ?
Solution :
Each seismograph gives us a locus that is a circle.
So, we have
Circle A has center (-5, 5) and radius 4
Circle B has center (-4, -3.5) and radius 5
Circle C has center (1, 1.5) and radius 7
Draw the three circles in a coordinate plane. The point of intersection of the three circles is the epicenter.
In the diagram above, three circles intersect at (-6, 1).
The epicenter is at about (-6, 1).
After having gone through the stuff given above, we hope that the students would have understood, "Locus".
Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.
WORD PROBLEMS
HCF and LCM word problems
Word problems on simple equations
Word problems on linear equations
Word problems on quadratic equations
Area and perimeter word problems
Word problems on direct variation and inverse variation
Word problems on comparing rates
Converting customary units word problems
Converting metric units word problems
Word problems on simple interest
Word problems on compound interest
Word problems on types of angles
Complementary and supplementary angles word problems
Markup and markdown word problems
Word problems on mixed fractrions
One step equation word problems
Linear inequalities word problems
Ratio and proportion word problems
Word problems on sets and venn diagrams
Pythagorean theorem word problems
Percent of a number word problems
Word problems on constant speed
Word problems on average speed
Word problems on sum of the angles of a triangle is 180 degree
OTHER TOPICS
Time, speed and distance shortcuts
Ratio and proportion shortcuts
Domain and range of rational functions
Domain and range of rational functions with holes
Graphing rational functions with holes
Converting repeating decimals in to fractions
Decimal representation of rational numbers
Finding square root using long division
L.C.M method to solve time and work problems
Translating the word problems in to algebraic expressions
Remainder when 2 power 256 is divided by 17
Remainder when 17 power 23 is divided by 16
Sum of all three digit numbers divisible by 6
Sum of all three digit numbers divisible by 7
Sum of all three digit numbers divisible by 8
Sum of all three digit numbers formed using 1, 3, 4
Sum of all three four digit numbers formed with non zero digits
