The points with the coordinates (2a,3a);(3b,2b);(c,c) are collinear if they lie on the same straight line. To determine whether these points are collinear or not, we can use the concept of slope.

Calculating the Slope:
The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)

Using the Slope Formula:
Let's calculate the slopes between the points (2a,3a) and (3b,2b), and between (3b,2b) and (c,c).

Slope between (2a,3a) and (3b,2b):
m1 = (2b - 3a) / (3b - 2a)

Slope between (3b,2b) and (c,c):
m2 = (c - 2b) / (c - 3b)

Checking for Collinearity:
If the slopes m1 and m2 are equal, then the three points are collinear. So, we need to check whether m1 = m2.

(2b - 3a) / (3b - 2a) = (c - 2b) / (c - 3b)

Simplifying the Equation:
To simplify the equation, we can cross-multiply and rearrange the terms to get:

(2b - 3a)(c - 3b) = (3b - 2a)(c - 2b)

Expanding and Simplifying:
Multiplying the terms on both sides, we get:

2bc - 6b^2 - 3ac + 9ab = 3bc - 6b^2 - 2ac + 4ab

Simplifying further:

2bc - 6b^2 - 3ac + 9ab = 3bc - 6b^2 - 2ac + 4ab

Canceling out Terms:
We can cancel out the common terms on both sides of the equation:

2bc - 3ac + 9ab = 3bc - 2ac + 4ab

Combining Like Terms:
Combining like terms, we get:

-3ac + 9ab = bc - 2ac + 4ab

Further Simplification:
Rearranging the terms, we get:

3bc - 4ab = 2ac - 9ab

Final Conclusion:
If the equation 3bc - 4ab = 2ac - 9ab holds true, then the points (2a,3a), (3b,2b), and (c,c) are collinear. If the equation does not hold true, the points are not collinear.

Summary:
To determine if the points (2a,3a), (3b,2b), and (c,c) are collinear, we calculated the slopes between the points and set them equal to each other. Simplifying the equation, we found that if 3bc