Find locus of centroid of a triangle AOB if line AB passes from (3,2) , A and B are on coordinate axes

To find the locus of the centroid of triangle AOB, where A and B are points on the coordinate axes and line AB passes through the point (3, 2), we need to establish some relationships first. The centroid of a triangle is the average of its vertices' coordinates. In this case, we can denote the coordinates of points A and B as A(0, a) and B(b, 0), where 'a' and 'b' are the y-coordinate of A and x-coordinate of B, respectively.

Understanding the Triangle's Points

Since points A and B lie on the axes, we can express their coordinates clearly:

• A = (0, a)
• B = (b, 0)

Finding the Equation of Line AB

The line AB can be described by the equation that connects these two points. The slope of line AB is calculated as follows:

The slope (m) of line AB is given by:

m = (0 - a) / (b - 0) = -a/b.

Using point-slope form from point A(0, a), the equation of line AB is:

y - a = (-a/b)(x - 0), which simplifies to y = -a/b * x + a.

Substituting the Point (3, 2)

Since line AB passes through the point (3, 2), we substitute these values into the line equation:

2 = -a/b * 3 + a.

Rearranging gives:

2 = a(1 - 3/b).

This implies:

a = 2b / (b - 3).

Finding the Centroid G

The centroid G of triangle AOB can be found using the formula:

G = ((x₁ + x₂ + x₃) / 3, (y₁ + y₂ + y₃) / 3).

For triangle AOB, the coordinates of the centroid will be:

G = ((0 + b + 0) / 3, (a + 0 + 0) / 3) = (b/3, a/3).

Substituting 'a' into the Centroid's Coordinates

Now, substituting the expression we found for 'a' into the centroid coordinates:

G = (b/3, (2b / (b - 3)) / 3) = (b/3, 2b / (3(b - 3))).

Eliminating 'b' to Find the Locus

To find the locus, we will eliminate 'b' between these two equations. Let’s denote the coordinates of the centroid as (x, y):

• x = b/3
• y = 2b / (3(b - 3)).

From the first equation, we can express b as:

b = 3x.

Substituting this back into the equation for y gives:

y = 2(3x) / (3(3x - 3)) = 6x / (9x - 9) = 2x / (3x - 3).

Clearing the fraction by multiplying both sides by (3x - 3) yields:

y(3x - 3) = 2x.

Rearranging this leads us to:

3xy - 3y = 2x.

Thus, we can express this as:

3xy - 2x - 3y = 0.

The Final Equation of the Locus

The final equation, which describes the locus of the centroid G of triangle AOB, is:

3xy - 2x - 3y = 0.

This equation represents a hyperbola, which shows how the centroid moves as points A and B change along their respective axes, keeping line AB passing through (3, 2).