# LOCUS math capsule

How to learn locus ?
What is a 'locus'?

## LOCUS

The curve described by any point which moves under the given geometrical conditions is called Locus.
e.g. The locus of a point (in a plane) Equidistant from a fixed point (in a plane) is a circle with the fixed point as centre.
The plural of locus is loci and it is a read as losai.
* The locus may be a line or lines , straight or curve.
* Every point which satisfies the given geometrical conditions lies on the locus and vice versa.
* A point which does not satisfy the given geometrical conditions cannot lie on the locus and vice versa.

### LOCUS IN SOME STANDARD CASES

1. LOCUS OF A POINT BETWEEN TWO PARALLEL LINES
The locus of a point which is equidistant from two parallel straight lines is a straight line parallel to the given lines and midway between them.
 LOCUS FIG 1

2. LOCUS OF A POINT PARALLEL TO A GIVEN LINE.
The locus of a point which is at a given distance from a given straight line, consists of a pair of straight lines parallel to the given line and at a given distance from it.

 LOCUS FIG 2

3. LOCUS OF A POINT AS THE CENTRE OF A WHEEL
The locus of the centre of a wheel which moves on a straight horizontal road is a straight line parallel to the road and at a distance equal to the radius of the wheel from the road.

 LOCUS FIG 3
4. LOCUS OF A POINT AS THE MID POINT OF PARALLEL CHORDS
The locus of the mid point of all parallel chords of a circle is the diameter of the circle which is perpendicular to the given parallel chords.
 LOCUS FIG 4
5. LOCUS OF A POINT EQUIDISTANT FROM THE TWO POINTS ON THE CIRCLE
The locus of a point which is inside a circle and is equidistant from two points on the circle is the diameter of the circle which is perpendicular to the chord of the circle joining the given point.

 LOCUS FIG 5

6. LOCUS OF A POINT OF INTERSECTION OF TWO SIDES OF A RIGHT ANGLE TRIANGLE CONTAINING RIGHT ANGLE
If A, B are fixed points then the locus of a point P such that ∠APB = 90 degree is the circle with AB as diameter .

 LOCUS FIG 6

7. LOCUS OF A POINT BETWEEN TWO CONCENTRIC CIRCLES
The locus of a point which is equidistant from a given circle consists of a pair of circles concentric with the given circle.

 LOCUS FIG 7
The locus of the centres of all circles passing through two given points A and B is the perpendicular bisector of the common chord AB.

Visuals or pictures makes it easier to learn about concepts ! isn't they ?

Start turn our attention to some Theorems based on loci.

## SOME THEOREMS BASED ON LOCI

* The loci of a point which is equidistant from two fixed points is perpendicular bisector of the line segment joining the two points.
 LOCUS FIG 8

* Here A, B are fixed points then all points on line 'l' i.e. PM are equidistant from both A and B . Then, PM⟂AB.
 LOCUS FIG 9
*The locus of a point equidistant from two intersecting lines is pair of lines bisecting the angles formed by the given lines.
 LOCUS FIG 10
Here AB, CD are two lines intersecting at ) forming the four angles . The line X'OX, Y'OY are bisecting these angles, then every point like P such that PN=PM lies on X'OX or Y'OY.
*The locus of the centre of all centre of all circles passing through two given points is the perpendicular bisector of the line segment.

*The locus of points which are equidistant from the three given non collinear points , is the centre of circle passing through these three non collinear points.
 LOCUS FIG 11
*The locus of the centre touching a given line AB at a given point C on it , is the straight line perpendicular to AB and C.
 LOCUS FIG 12
*There is no locus of points which are equidistant from three distinct points on a line . As any point equidistant from A, B and C must be common to both.
* In ∠BAC, the locus of a point which lies in the interior of ∠BAC and equidistant from two lines AB and AC is the bisector of ∠BAC.
 LOCUS FIG 13

Nice blog post , Isn't it?
Now, Lets do some exercises together , friend.

## EXERCISES ON LOCUS:

 LOCUS MATH CAPSULE 1

 LOCUS MATH CAPSULE 2

 LOCUS MATH CAPSULE 3

 LOCUSMATH CAPSULE 4
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