Question

Find the equation of a locus of moving point such that the slope of line joining the point to A(1,3) is three times that of the slope of the line joining the point to B(3,1)

Answer 1

Find the equation of a locus of moving point such that the slope of line joining the point to $A(1,3)$ is three times that of the slope of the line joining the point to $B(3,1)$

Solution:

A locus describes a set of points $P(x, y)$ that obeys certain conditions, or a single point $P(x, y)$ that moves along a certain path.

If a point moves on a plane satisfying some given geometrical condition then the path trace out by the point in the plane is called its locus. By definition, a locus is determined if some geometrical condition are given. Evidently, the co-ordinate of all points on the locus will satisfy the given geometrical condition. The algebraic form of the given geometrical condition which is satisfied by the co-ordinate of all points on the locus is called the equation to the locus of the moving point. Thus, the co-ordinates of all points on the locus satisfy its equation of locus: but the co-ordinates of a point which does not lie on the locus do not satisfy the equation of locus. Conversely, the points whose co-ordinates satisfy the equation of locus lie on the locus of the moving point.

Let the moving point be $(x,y)$. Slope of moving point with respect to point $A(1,3) = \frac{(y - 3)}{(x - 1)}$. Slope of the moving point with respect to point $B(3,1) = \frac{(y - 1)}{(x - 3)}$.

We are given that the slope of the line joining $(x, y)$ with point $A(1, 3)$ is three times that of the slope of the line joining the point to $B(3, 1)$ therefore our equation becomes:

$$\frac{(y - 3)}{(x - 1)} = \frac{3(y - 1)}{(x - 3)}$$

Simplify our equation: $3(y - 1)(x - 1) = (y - 3)(x - 3)$

$$3(xy - y - x + 1) = (xy - 3y - 3x + 9)$$ $$3xy - 3x - 3y + 3 = xy - 3y - 3x + 9$$ $$3xy - 3x - 3y + 3 - xy + 3x + 3y - 9 = 0$$ $$2xy - 6 = 0$$

So, we find the equation of a locus of moving point: $y = \frac{3}{x}$

The equation of a locus of moving point can be represented graphically.