Animated Solution for Mathematics - Straight Lines: A point P moves on the line $2x-3y+4=0$. If Q(1, 4) and R(3, -2) are fixed points, then the locus of the centroid of $\Delta PQR$ is a line :

Imagine a point $P(\alpha ,\beta )$ dancing along the line $2x-3y+4=0$. We are given two fixed observers, $Q(1,4)$ and $R(3,-2)$.

As $P$ moves, it forms a triangle $△PQR$. The centroid $G(h,k)$ acts as the balance point of this ever-changing triangle. Our goal is to determine the locus of $G$.

The centroid formula serves as our compass. It states that the coordinates of the centroid are the arithmetic mean of the vertices' coordinates.

Given $P(\alpha ,\beta )$, $Q(1,4)$, and $R(3,-2)$, the centroid $G(h,k)$ is defined by:

To find the locus, we must express the coordinates of $P$ in terms of the coordinates of $G$. Rearranging the centroid equations, we obtain:

This transformation acts as the key that unlocks the relationship between the moving point $P$ and the resulting centroid $G$.

The point $P$ is constrained to lie on the line $2x-3y+4=0$. Therefore, the coordinates $(\alpha ,\beta )$ must satisfy the equation:

Substituting our expressions for $\alpha$ and $\beta$ into this constraint, we get:

Expanding the equation above, we perform the following algebraic steps:

Replacing $(h,k)$ with the general coordinates $(x,y)$, we find that the locus of the centroid is:

The slope of this line is $\frac{2}{3}$, which is identical to the slope of the original line. This confirms that the centroid transformation preserves the orientation of the path, resulting in a line parallel to the original.