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: Use the line equation:
If $C$ lies on line $AB$, then $C = A + t(B-A)$ for some number $t$. We find $t$ from the $x$-coordinate and test the others.

Step 2: Solve from the x-coordinate:
We get $1 = 6 + 10t$, so $t = -\frac{1}{2}$.

Step 3: Test y and z:
For $y$: $-7 + (-12)\left(-\frac12\right) = -7 + 6 = -1$. It matches.
For $z$: $0 + (-4)\left(-\frac12\right) = 2$. It matches.
So $C$ lies on the line $AB$ (outside the segment, since $t<0$). The Assertion is true.

Step 4: Test the Reason:
A mid-point needs $t = \frac12$. Here $t = -\frac12$. So $C$ is not the mid-point, yet the points are collinear. This shows the mid-point condition is not required for collinearity.
So the Reason is false.

Final Answer:
Assertion true, Reason false: option (C). \[ \boxed{\text{(C)}} \]