Locus - Practice Questions & MCQ

Locus and its Equation

When a point moves in a plane under certain geometrical conditions, then the point traces out a path, This path of the moving point is known as the locus of this point.   

For example, let a point O(0,0) is a fixed point (i.e. origin) and a variable point P (x, y) is in the same plane. If point P moves in such a way that the distance OP is constant r, then point P traces out a circle whose center is O(0, 0) and radius is r.

Steps to Finding the Equation of Locus

Consider the point (h, k) whose locus is to be found.

Express the given condition as an equation in terms of the known quantities and unknown parameters.

Eliminate the parameters so that the resultant equation consists only locus coordinates h, k, and known quantities.

Now, replace the locus coordinate (h, k) with (x, y) in the resultant equation.

Illustriation

Find the equation of the locus of the point which is at a constant distance of 5 units from a point $(2,3)$
Solution
Let $A=(2,3)$ and $B=(h, k)$
$\mathrm{AB}=$ constant $=5$
$(A B)^2=5^2=25$
$(A B)^2=(h-2)^2+(k-3)^2=25$

$
h^2-4 h+4+k^2-6 k+9=25
$

The equation of locus is

$
x^2+y^2-4 x-6 y-12=0
$
In next chapter, we will see that this equation represents circle with centre at the point $(2,3)$ with radius 5 unit