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Find the value of p for which the points (−3, 4), (p, 0) and (6, −8) are collinear.

Class 10Coordinate GeometryGeometry — Coordinates
  • A2
  • B-2
  • C6
  • D0

Step-by-Step Solution

Collinear means the triangle they form has zero area: ₁₂ − ₃) + ₂₃ − ₁) + ₃₁ − ₂) = 0 → → → → , so the middle point is the origin. All three terms must be included: dropping the first term gives and dropping the last gives , while 6 is just the -coordinate of the third point copied out.
Answer: D) 0

Key idea: Area of a triangle from vertices

Given A₁, ₁), B₂, ₂), C₃, ₃), the area is:

**Area = ₁₂ − ₃) + ₂₃ − ₁) + ₃₁ − ₂)|**

The absolute value sign |·| ensures area is positive regardless of vertex order.

**Use case 1**: Compute area without drawing anything.
**Use case 2**: Test for COLLINEARITY — three points are collinear iff the (signed) area expression equals 0.

**Example**: A(1, 1), B(2, 3), C(4, 5).
Area = ·2 = 1.
Non-zero → not collinear.

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