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A locus or curve is the set of points and only those points satisfying a given condition or a well defined property. This is the locus definition in geometry. Example of locus is provided below:

(1) A,B are two fixed points. Let a set of points equidistant from A and B. Locus is the perpendicular bisector of AB . Every point on the perpendicular bisector of AB obeys the condition.

(2) The locus of a set of points which are at a constant distance from a fixed point,is a circle.

Loci and Concurrency

  • A locus of points is the set of points, and only those points, that satisfy given conditions.

- Every point satisfying the given conditions lies on the locus.

- Every point on the locus satisfies the given conditions.

- The locus can be a straight line or a curved line (lines).

  • To determine a locus:

- State what is given and the condition to be satisfied.

- Find several points satisfying the condition, which indicate the shape of the locus.

- Connect the points and describe the locus fully.

  • The locus of points equidistant from the sides of a given angles is the bisector of the angle.


  • The locus of points equidistant from two given intersecting lines is the bisectors of the angles formed by the lines.

Locus Online

  • The locus of points equidistant from two fixed points is the perpendicular bisector of the segment joining them.

Locus Definition

  • Locus of a point equidistant from a given point in a plane is a circle.

Locus Math

  • The locus of points equidistant from two given parallel lines is a line parallel to the two lines and midway between them.

Locus of a Point

  • Locus of all points at a given distance from a given line is two straight lines parallel to the given line.

Locus equation:

The equation to a locus is the condition which the coordinates of each of the points of that locus and only those points must satisfy. In other words, the equation to a locus is nothing but the geometrical property expressed in algebraic language.

Conversely, let (x,y) represent any point on the locus, x and y satisfy the equation of the locus and every point satisfying the equation lies on the locus.

If the functional relation between x,y be f (x,y) = 0, then we say f (x,y ) = 0 is the Cartesian equation of the locus.

Thus in coordinate geometry a locus is represented by an equation. Let us see locus of equation in locus learning.


Equation of Locus

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The set of points (x,y) ,satisfying a given equation f (x,y) = 0 is called the locus or graph of the equation.

In other words, if f is a relation on R, then the graph of f is the set of all (x,y) `in` R, in a coordinate plane.

e.g., The set of a points which are at a constant distance of 5 units from a fixed point (2,3) has the equation

sqrt(( x -2 )2 + ( y - 3 )2) = 5

=> x2 + y2 - 4x - 6y - 12 = 0

Solved Problems on Locus

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Below are some solved problems on locus

Q 1: Find the locus of the point which is at a constant distance of 10 units from (2,-8) .

solution : P(x,y) `in` Locus. If A = (2,-8) , given geometrical condition is PA = 10

=> PA2 = 100

=> (x -2 2) + ( y+8 2) = 100

=>Equation of locus is x2 + y2 - 4x +16y - 32 = 0

Q 2: Find the equation to the set of points equidistant from (-1,-1)and (4,2).

solution : Let A =( -1,-1) and B =( 4,2)

If P( x,y ) `in ` set then the geometrical condition satisfied by P is


=> PA2 = PB2

=> ( x +1) 2 + ( y + 1 )2 = ( x - 4 )2 + (y - 2) 2

=> x2 +2x +1 +y2 +2y +1 = x2 - 8x + 16 + y2 - 4y + 4

=> 5x + 3y = 9

Therefore, the equation of locus is 5x + 3y = 9

Q 3: Find the locus of a point which moves in such a way that its distance from (3,0) is always equal to its distance from the origin.

solution : Let A = (3,0) and O = (0,0).

Let P( x,y ) be a point satisfying the geometrical condition.,


=> PA2 = OP2

=> ( x - 3 )2 + y2 = x2 +y2

=> 6x = 9

Therefore, the equation of locus of P is 2x - 3 = 0.

Q 4: If p is the length of perpendicular drawn from the origin on the line AB whose intercepts on the axes are a and b, then show


Equation of the line AB in the intercept form is

Since (i) and (ii) represent the same line, we have

Squaring and adding, we get

Q 5: A straight line passes through the point (a, b) and this bisects the part of the line intercepted between the axes.
Show that

Let the straight line passing through p(a, b) cut the x-axis at A(a,0) and B(0,b) on the x-axis and the y-axis respectively.

More topics in Locus
Axis of Symmetry
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