Coordinate System and Coordinates
15.0 Locus
15.0 Locus
The locus of a moving point is the path traced out by that point under one or more given conditions. It may also be defined as a set of points whose location satisfies or is determined by one or more specified conditions.
Steps to find the locus of a point
- Assume $(h,k)$ to be the coordinates of moving point say $P$.
- Apply geometrical conditions on $h,k$ which gives a relation between $h$ and $k$.
- Now replace $h$ by $x$ and $k$ by $y$ in the eliminant and resulting equation would be the equation of locus.
Question 6. Find the locus of the moving point $P$ such that $2PA=3PB$, where $A$ is $(0,0)$ and $B$ is $(4,-3)$.
Solution: Let $P(h,k)$ be the moving point whose locus is required. It is given that $$2PA = 3PB$$
Squaring both sides we get,
$$\begin{equation} \begin{aligned} 4P{A^2} = 9P{B^2} \\ 4({h^2} + {k^2}) = 9\{ {(h - 4)^2} + {(k + 3)^2}\} \\ 4({h^2} + {k^2}) = 9({h^2} + {k^2} - 8h + 6k + 25) \\ 5{h^2} + 5{k^2} - 72h + 54k + 225 = 0 \\\end{aligned} \end{equation} $$
Changing $(h,k)$ to $(x,y)$ then, $$5{x^2} + 5{y^2} - 72x + 54y + 225 = 0$$ which is the required locus of $P$.
