Question 4 of 25intermediate🔧 ApplyShort Answer4 marks

Show that the points (2,3,4),(−1,−2,1),(5,8,7)(2, 3, 4), (-1, -2, 1), (5, 8, 7) are collinear.

Correct Answer

Let A = (2, 3, 4), B = (-1, -2, 1), C = (5, 8, 7).

Direction ratios of AB: -1 - 2 = -3, -2 - 3 = -5, 1 - 4 = -3 So, direction ratios of AB are -3, -5, -3.

Direction ratios of BC: 5 - (-1) = 6, 8 - (-2) = 10, 7 - 1 = 6 So, direction ratios of BC are 6, 10, 6.

Checking proportionality: -3/6 = -1/2, -5/10 = -1/2, -3/6 = -1/2

The direction ratios of AB and BC are proportional, hence AB is parallel to BC. But point B is common to both AB and BC. Therefore, A, B, C are collinear points.

Exercise: EXERCISE 11.1 | Q: 4 | (Page Page 5)
For More Understanding

Explanation

Following the method shown in Example 5 of the textbook, we prove collinearity by showing that direction ratios of consecutive line segments are proportional. When direction ratios are proportional, the lines are parallel. If two lines share a common point and are parallel, they must lie on the same straight line, making the points collinear.

Solution Steps

  1. Step 1: Label the points as A(2,3,4), B(-1,-2,1), C(5,8,7)

  2. Step 2: Find direction ratios of AB: (-3, -5, -3)

  3. Step 3: Find direction ratios of BC: (6, 10, 6)

  4. Step 4: Verify proportionality: all ratios equal -1/2

  5. Step 5: Conclude collinearity since AB ∥ BC with common point B

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