Question

If the points \( A(3,4) \), \( B(x_1,y_1) \) and \( C(x_2,y_2) \) are such that both \( 3,x_1,x_2 \) and \( 4,y_1,y_2 \) are in A.P., then

• A. \( A, B, C \) are vertices of an isosceles triangle
• B. \( A, B, C \) are collinear points
• C. \( A, B, C \) are vertices of a right angled triangle
• D. \( A, B, C \) are vertices of a scalene triangle
• E. \( A, B, C \) are vertices of an equilateral triangle

Hint:

If coordinates themselves follow A.P., try expressing one point in terms of another — it often leads to collinearity.

Answer 1

The Correct Option is B

Concept:
• If three numbers are in A.P., then the middle term is the average of the other two: \[ b = \frac{a + c}{2} \]
• Collinearity condition: Three points are collinear if slopes between pairs are equal.
• Slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Step 1: Using A.P. condition for \(x\)-coordinates.
Given \( 3, x_1, x_2 \) are in A.P.: \[ x_1 = \frac{3 + x_2}{2} \] Rewriting: \[ 2x_1 = 3 + x_2 \quad \Rightarrow \quad x_2 = 2x_1 - 3 \quad \cdots (1) \]

Step 2: Using A.P. condition for \(y\)-coordinates.
Given \( 4, y_1, y_2 \) are in A.P.: \[ y_1 = \frac{4 + y_2}{2} \] Rewriting: \[ 2y_1 = 4 + y_2 \quad \Rightarrow \quad y_2 = 2y_1 - 4 \quad \cdots (2) \]

Step 3: Writing coordinates of points.
\[ A = (3,4), \quad B = (x_1,y_1), \quad C = (2x_1 - 3, 2y_1 - 4) \]

Step 4: Finding slope of \(AB\).
\[ m_{AB} = \frac{y_1 - 4}{x_1 - 3} \]

Step 5: Finding slope of \(BC\).
\[ m_{BC} = \frac{(2y_1 - 4) - y_1}{(2x_1 - 3) - x_1} \] \[ m_{BC} = \frac{y_1 - 4}{x_1 - 3} \]

Step 6: Comparing slopes.
\[ m_{AB} = m_{BC} \] Since slopes are equal, points lie on the same straight line.

Step 7: Conclusion.
Thus, \( A, B, C \) are collinear points.