Question

Assertion (A) : Points \(A(6, -7, 0)\), \(B(16, -19, -4)\) and \(C(1, -1, 2)\) are collinear.

Reason (R) : Three points \(A\), \(B\) and \(C\) are collinear, when point \(C\) is the mid-point of line segment joining points \(A\) and \(B\).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

Here \(\overrightarrow{AB} = (10,-12,-4)\) and \(\overrightarrow{AC} = (-5,6,2)\) are parallel, so (A) is true; \(C\) is not the mid-point, so (R) is false.

Answer 1

The Correct Option is C

Step 1: Plan:
Three points are collinear when the vectors \(\overrightarrow{AB}\) and \(\overrightarrow{AC}\) are parallel. We check the Assertion by finding these vectors.

Step 2: Find the two vectors:
\[ \overrightarrow{AB} = (16-6,\ -19+7,\ -4-0) = (10,\ -12,\ -4) \]
\[ \overrightarrow{AC} = (1-6,\ -1+7,\ 2-0) = (-5,\ 6,\ 2) \]

Step 3: Compare them:
We see \(\overrightarrow{AB} = -2\,\overrightarrow{AC}\), because \(10=-2(-5)\), \(-12=-2(6)\) and \(-4=-2(2)\). So the vectors are parallel and share the point \(A\). The points are collinear. Assertion (A) is TRUE.

Step 4: Check the Reason (R):
The mid-point of \(AB\) is \(\left(\frac{6+16}{2},\frac{-7-19}{2},\frac{0-4}{2}\right) = (11,-13,-2)\). This is not \(C(1,-1,2)\).
Also, collinear points need not have the middle point at the exact centre. Any point on the line works. So the Reason, as a condition for collinearity, is FALSE.

Step 5: Pick the code:
(A) is true and (R) is false. That is code (C).

Final Answer:
Option (C) is correct. \[ \boxed{\text{(A) true, (R) false}} \]