What is a LOCUS?? Is a perpendicular bisector a LOCUS??

Give some egs. of a Locus:-

#Geometry #Locus #Circles

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Comments

There are two useful ways to interpret 'locus'

Locus is a set of points that satisfy some given condition or property.

There are some conditions (in geometry) which provide definite position of a point.

A point which is 2 units from origin and lies on the positive x-axis is $2,0$ .

But sometimes the conditions are less definite.

A point which is 2 units from origin and lies on the x-axis is not a single point. There are two such points: $-2,0$ & $2,0$

Now consider the condition: Points which lie at 2 units from the origin.This condition is less definite in the sense that there isn't only a single point or only two such points. How many such points are there? Infinite. But the locus, what is the locus? How are those points arranged? What is the position of all those points?All the points lie on the circle of radius 2 units and center at the origin. The locus is a circle.

A second interpretation which does come handy while solving problem is this: Locus is the path or curve traced by a moving point satisfying the given condition. This description clearly points out that there is no locus of a fixed point.

What is the locus of the Point which lies at 2 units from the origin? The point concerned in not fixed. It is variable. To find the locus means to find 'how it can vary?' or what path the variable point traces?

Some e.g of locus are:

Locus of a point which lies at a given distance from a given point is a circle.

Locus of a point which lies at a given distances from a given line is a parallel line.

Locus of a point which lie at equal distance from two given points is the perpendicular bisector of the line joining the two given points.

Soumo Mukherjee - 6 years, 1 month ago

Thanks buddy :-)

Sunny Sharma - 5 years, 1 month ago

A locus is simply a set of points that satisfy some conditions.

The perpendicular bi-sector of a line segment is definitely a locus since it is the set of points that are equidistant from both the end-points of the segment.

A circle is another example of a locus. It is the set of points that are a certain distance (called the radius of the circle) away from a certain point (called the center of the circle).

Hope this helps!

Mursalin Habib - 6 years, 1 month ago

Thanks buddy :-))

Sunny Sharma - 5 years, 1 month ago

Locus? Hmm.. Well I can relate to what you are feeling, felt the same when encountered the term 'locus'.

A locus basically is a collection (set) of points that satisfy some give condition(s), like if someone asks you how many points can you plot on the Cartesian plane which are at a distance of 5 units away from the origin? In the beginning you would start thinking in the wrong direction.(which is quite important, 'coz only then will you be able to see the beauty of Math)You might try to count the points but you won't get far, like you might find (5,0), (-5,0),(3,4) , (-3,4) etc...But after some time you will realize that there are points which you cant resolve easily.

Then there will be a click in your brain and you will remember something you studied in your Geometry class. In a circle, all the points on the circumference lie at the same distance from the center of the circle..and then the realization sets in that there are infinite points that can be at a distance of 5 units from the origin. You ask how? I say, draw a circle of radius 5 units centered at origin and all the points on its circumference will be at a distance of 5 units from the origin...(Try to derive the equation of the locus.Remember, start from the basics(use Pythagoras theorem))

Now back to your question..Is perpendicular bisector of a line a locus??

Yes it is. A perpendicular bisector bisects a line perpendicularly i.e. all the points on a perpendicular bisector satisfy the condition that they all are equidistant from the two end points of the line segment.So we have a condition, and we have points that satisfy it, thus it is a locus.

Some other examples-

Locus of points that satisfy the condition,

1) equidistant from both x and y- axes: is the line y=x (LOL)

2) at a given distance from a given point is a circle.

3)equidistant from a line and some other point is a parabola(Don't fret, you will be taught this later)

So on, and so forth.

Hope this helps you.Feel free to ask any further doubts.

Pankaj Joshi - 6 years, 1 month ago

Now its clear to me thanks buddy :-))

Sunny Sharma - 5 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$