Locus
Locus in Coordinate Geometry:
Definition:
 Concept: A locus represents a set of points that satisfy a particular condition or property.
 In Coordinate Geometry: The locus is described as the path traced by a point (or points) that satisfy a given condition.
Representation:
 Equation: The locus can often be represented by an equation involving the coordinates $x$ and $y$ that expresses the relationship between them.
Types of Loci:

Linear Loci:
 Examples: Lines, rays, line segments.
 Equation: For example, the equation of a line $y=mx+c$ represents a line's locus.

Circular Loci:
 Examples: Circles, arcs.
 Equation: For instance, the equation ${x}^{2}+{y}^{2}={r}^{2}$ represents a circle's locus with radius $r$ centered at the origin.

Conic Sections:
 Examples: Ellipses, parabolas, hyperbolas.
 Equations: Equations like $\frac{{x}^{2}}{{a}^{2}}\frac{{y}^{2}}{{b}^{2}}=1$ represent hyperbolic loci.

Parametric Equations:
 Definition: Loci can be represented using parametric equations, involving additional parameters.
 Example: Parametric equations like $x=\mathrm{cos}(t)$, $y=\mathrm{sin}(t)$ represent a locus of a point moving along a unit circle as $t$varies.
Application:
 Problem Solving: Helps in solving geometric problems involving the path or trajectory of points.
 Curve Sketching: Useful for visualizing and understanding shapes described by equations.
Example:
Find the locus of points that are equidistant from the points (3, 4) and (5, 2).
Steps to Solve:

Distance Formula:
 Consider a point P(x, y) equidistant from (3, 4) and (5, 2).
 Use the distance formula to set up equations for the distances between P and each given point.

Equation Setting:
 Distance between point P(x, y) and (3, 4): $\sqrt{(x(3){)}^{2}+(y4{)}^{2}}$
 Distance between point P(x, y) and (5, 2): $\sqrt{(x5{)}^{2}+(y(2){)}^{2}}$

 Equate these distances to find the locus.

Solve for Locus:
 Set the two equations equal to each other and solve for the locus equation in terms of x and y.
Solution:
Let's equate the distances to find the locus equation:
$\sqrt{(x+3{)}^{2}+(y4{)}^{2}}=\sqrt{(x5{)}^{2}+(y+2{)}^{2}}$
Squaring both sides to eliminate square roots:
$(x+3{)}^{2}+(y4{)}^{2}=(x5{)}^{2}+(y+2{)}^{2}$
Expanding and simplifying the equation:
${x}^{2}+6x+9+{y}^{2}8y+16={x}^{2}10x+25+{y}^{2}+4y+4$
Simplifying further:
$6x10x+925+{y}^{2}8y4y+164=0$
$4x12y4=0$
Finally, rearranging the equation to isolate the locus:
$x+3y+1=0$
Interpretation:
The locus of points equidistant from (3, 4) and (5, 2) is represented by the equation $x+3y+1=0$ in the coordinate plane. This equation describes a straight line passing through the points (3, 4) and (5, 2), illustrating the path traced by points equidistant from these two given points.