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The points with coordinates (2a,3a), (3b...

The points with coordinates `(2a,3a), (3b,2b)` and `(c,c)` are collinear

A

for no value of a, b, c

B

for all values of a, b, c

C

if a, `(c )/(5)`, b are in HP

D

if `a, (2c)/(5), b` are in HP

Text Solution

AI Generated Solution

The correct Answer is:
To determine if the points \((2a, 3a)\), \((3b, 2b)\), and \((c, c)\) are collinear, we can use the concept of slopes. If the slopes between each pair of points are equal, then the points are collinear. ### Step-by-Step Solution: 1. **Identify the Points**: The points given are: - Point 1: \( P_1 = (2a, 3a) \) - Point 2: \( P_2 = (3b, 2b) \) - Point 3: \( P_3 = (c, c) \) 2. **Calculate the Slope Between \(P_1\) and \(P_2\)**: The slope \(m_{12}\) between points \(P_1\) and \(P_2\) is given by: \[ m_{12} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2b - 3a}{3b - 2a} \] 3. **Calculate the Slope Between \(P_2\) and \(P_3\)**: The slope \(m_{23}\) between points \(P_2\) and \(P_3\) is given by: \[ m_{23} = \frac{c - 2b}{c - 3b} \] 4. **Set the Slopes Equal**: For the points to be collinear, the slopes must be equal: \[ \frac{2b - 3a}{3b - 2a} = \frac{c - 2b}{c - 3b} \] 5. **Cross Multiply**: Cross multiplying gives us: \[ (2b - 3a)(c - 3b) = (c - 2b)(3b - 2a) \] 6. **Expand Both Sides**: Expanding the left side: \[ 2bc - 6b^2 - 3ac + 9ab \] Expanding the right side: \[ 3bc - 2ac - 6b^2 + 4ab \] 7. **Combine Like Terms**: Rearranging gives us: \[ 2bc - 3ac + 9ab - 6b^2 = 3bc - 2ac - 6b^2 + 4ab \] Simplifying this results in: \[ 2bc - 3ac + 9ab = 3bc - 2ac + 4ab \] 8. **Rearranging Terms**: Bringing all terms to one side: \[ 2bc - 3bc + 2ac + 9ab - 4ab = 0 \] This simplifies to: \[ -bc + 2ac + 5ab = 0 \] 9. **Rearranging for \(c\)**: Rearranging gives: \[ 5ab = bc - 2ac \] Dividing through by \(abc\) (assuming \(a, b, c \neq 0\)): \[ 5 = \frac{1}{a} + \frac{1}{b} + \frac{2}{c} \] 10. **Conclusion**: The relationship derived indicates that the points are collinear if the condition derived holds true.
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