Introduction To Locus


I. Introduction
A locus is defined as a set of all points satisfying one or more given conditions. As per some older definitions, locus was considered as the path of a point moving following some condition.
Some examples are as follows:
In two dimension, the set of points that are equidistant from two fixed points, form a line, which is the perpendicular bisector of the line segment joining the two fixed points. In three dimension, the same condition will give us a set of points that form a plane.

Similarly, in two dimension, the set of points at a constant distance from another fixed point gives us a circle. In three dimension the same rule gives us a spherical surface.

All the conic sections, ellipse, hyperbola, parabola can be defined as locii in more than one way.
For example,
Ellipse can be defined as the locus of a point, the sum of whose distances from two fixed points is constant.
Ellipse can also be defined as the locus of a point, such that the ratio of its distances from a fixed point and fixed line (not passing through the fixed point), is in a fixed ratio of less than $1$.

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II. Common Loci 
Locus of a point dividing a line segment joining a point $P$ moving along the circumference of a circle and a fixed point $A$, in the ratio $m:n$ 

Let $P (x_p, y_p)$ be a moving point on the circle represented by $x^2 + y^2 = r^2$, and $A (a, b)$ be any fixed point on the plane.
Let $L (x_L, y_L)$ be the point on the line segment $AP$, dividing it in the ratio $m:n$

Since $L$ divides $AP$ in the ration $m:n$:
$x_L = \dfrac{na + mx_p}{m + n}$

$\Rightarrow x_p = \dfrac{x_L(m + n) - na}{m}$

and
$y_L = \dfrac{nb + my_p}{m + n}$

$\Rightarrow y_p = \dfrac{y_L(m + n) - nb}{m}$

Since $x_p, y_p$ lie on the given circle:
${x_p}^2 + {y_p}^2 = r^2$

$\therefore \left[\dfrac{x_L(m + n) - na}{m} \right]^2 + \left[\dfrac{y_L(m + n) - nb}{m} \right]^2 = r^2$

$\Rightarrow [x_L(m + n) - na]^2 + [y_L(m + n) - nb]^2 = m^2r^2$

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$\Rightarrow \left(x_L - \dfrac{na}{m + n} \right)^2 + \left(y_L - \dfrac{nb}{m + n}  \right)^2 = \dfrac{m^2r^2}{(m + n)^2}$

$\Rightarrow \left(x_L - \dfrac{na}{m + n} \right)^2 + \left(y_L - \dfrac{nb}{m + n}  \right)^2 = \left( \dfrac{mr}{m + n} \right)^2$

Therefore, the locus of the point $L (x_L, y_L)$, is a circle centered at $\left( \dfrac{na}{m + n} , \dfrac{nb}{m + n} \right)$ with a radius of $\dfrac{mr}{m + n}$ .

The following widget helps you to develop a better understanding of this locus. Try varying the size of the circle by moving the point $R$ (blue dot), and moving the point $A$ anywhere (inside or outside the circle), and the point $L$ to change the ratio of the two segments. Then move the point $P$ along the circumference of the circle to view the locus of the point $L$. 

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