Question:
If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :
If points (2, 3), (0, 4) and (p, 2) are collinear, then the value of p is :
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For checking collinearity of three points, the slope method is usually the fastest.
If one x-coordinate is zero (like in this question), calculations become significantly simpler.
If one x-coordinate is zero (like in this question), calculations become significantly simpler.
Updated On: Sep 10, 2026
- \(\frac{4}{7}\)
- \(-\frac{3}{7}\)
- \(4\)
- \(-4\)
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The Correct Option is C
Solution and Explanation
Concept:
• Three points \(A(x_1, y_1), B(x_2, y_2)\) and \(C(x_3, y_3)\) are collinear if they lie on the same straight line.
• This condition is met if the slope of line segment \(AB\) is equal to the slope of line segment \(BC\).
• Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Step 1: Calculate the slope of the segment joining (2, 3) and (0, 4)
Let \(A = (2, 3)\) and \(B = (0, 4)\).
Slope of \(AB\) (\(m_1\)) is:
\[ m_1 = \frac{4 - 3}{0 - 2} = \frac{1}{-2} = -\frac{1}{2} \]
Step 2: Calculate the slope of the segment joining (0, 4) and (p, 2)
Let \(B = (0, 4)\) and \(C = (p, 2)\).
Slope of \(BC\) (\(m_2\)) is:
\[ m_2 = \frac{2 - 4}{p - 0} = \frac{-2}{p} \]
Step 3: Equate the slopes to find p
Since the points are collinear, \(m_1 = m_2\):
\[ -\frac{1}{2} = -\frac{2}{p} \] Cancel the negative signs from both sides:
\[ \frac{1}{2} = \frac{2}{p} \] Cross-multiplying:
\[ p \times 1 = 2 \times 2 \] \[ p = 4 \] Alternatively, using the area of triangle method, set \(\frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| = 0\):
\[ |2(4-2) + 0(2-3) + p(3-4)| = 0 \] \[ |2(2) + 0 + p(-1)| = 0 \] \[ |4 - p| = 0 \implies p = 4 \] This matches option (C).
• Three points \(A(x_1, y_1), B(x_2, y_2)\) and \(C(x_3, y_3)\) are collinear if they lie on the same straight line.
• This condition is met if the slope of line segment \(AB\) is equal to the slope of line segment \(BC\).
• Slope formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
Step 1: Calculate the slope of the segment joining (2, 3) and (0, 4)
Let \(A = (2, 3)\) and \(B = (0, 4)\).
Slope of \(AB\) (\(m_1\)) is:
\[ m_1 = \frac{4 - 3}{0 - 2} = \frac{1}{-2} = -\frac{1}{2} \]
Step 2: Calculate the slope of the segment joining (0, 4) and (p, 2)
Let \(B = (0, 4)\) and \(C = (p, 2)\).
Slope of \(BC\) (\(m_2\)) is:
\[ m_2 = \frac{2 - 4}{p - 0} = \frac{-2}{p} \]
Step 3: Equate the slopes to find p
Since the points are collinear, \(m_1 = m_2\):
\[ -\frac{1}{2} = -\frac{2}{p} \] Cancel the negative signs from both sides:
\[ \frac{1}{2} = \frac{2}{p} \] Cross-multiplying:
\[ p \times 1 = 2 \times 2 \] \[ p = 4 \] Alternatively, using the area of triangle method, set \(\frac{1}{2}|x_1(y_2-y_3) + x_2(y_3-y_1) + x_3(y_1-y_2)| = 0\):
\[ |2(4-2) + 0(2-3) + p(3-4)| = 0 \] \[ |2(2) + 0 + p(-1)| = 0 \] \[ |4 - p| = 0 \implies p = 4 \] This matches option (C).
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