Q.The equation of the locus of the point whose distance from the $y$-axis is half the distance from the origin is

A point is not a thing but a place — a position located, with reference to a coordinate system, by a unique ordered pair $(x,y)$: $x$ is the (signed) distance from the $y$-axis and $y$ the (signed) distance from the $x$-axis. The locus of a moving point is the path traced out by it as it moves subject to a stated geometric condition; equivalently, an equation in $x,y$ has infinitely many real solution-pairs $(x,y)$, and the collection of all the points corresponding to those solutions is the locus of the equation. (The plural of locus is loci.)

Some standard loci: a point moving so as to stay equidistant from two fixed points $A,B$ traces the perpendicular bisector of $AB$; a point equidistant from two fixed lines traces the angle bisector; a point at a fixed distance $r$ from a fixed point $O$ traces a circle of radius $r$ centred at $O$. A point on the rim of a circle rolling along a straight line traces a cycloid.

Procedure for finding the equation of a locus.

1. Assign coordinates $(h,k)$ to the moving point $P$ whose locus is wanted.
2. Translate the given geometric condition(s) into equation(s) relating $h,k$ and any parameters/known quantities.
3. Eliminate the parameter(s), so that only $h,k$ and known constants remain.
4. Replace $h$ by $x$ and $k$ by $y$; the resulting equation in $x,y$ is the equation of the locus.

Worked pattern. For $P$ equidistant from two fixed points $A(x_1,y_1)$ and $B(x_2,y_2)$: set $P=(h,k)$, write $PA=PB$, square both sides, and expand — the $h^2,k^2$ terms cancel, leaving a linear equation (the perpendicular bisector). For $P$ at a fixed distance from a fixed point, $PA=r$ squared gives the circle equation directly. For a parametric locus $(f(\theta),g(\theta))$, solve each coordinate equation for the trig function of $\theta$ it contains, then combine using an identity to eliminate $\theta$ entirely — this is exactly how the standard hyperbola/ellipse/astroid forms arise from their parametric descriptions.

The phrase "locus of a point class 11 maths definition and examples" shows up often in student searches tied to the Straight Lines chapter of the NCERT/CBSE Class 11 Mathematics curriculum, where locus problems are a recurring important-questions category in both board exams and JEE Main. Learning the standard four-step procedure here also pays off directly in later chapters on circles and conic sections that build on the same idea.