Question

For the line \(\frac{x+1}{1} = \frac{y-2}{2} = \frac{z+3}{3}\), identify the incorrect statement among the following.

• A. It can be represented by equation \(\frac{x+2}{1} = \frac{y}{2} = \frac{z+6}{3}\)
• B. It lies in the plane \(x-2y+z+8 = 0\)
• C. It is perpendicular to the plane \(x-2y+z = 0\)
• D. It passes through point \((0,4,0)\).

Hint:

Perpendicular to a plane means parallel to its normal.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The given line passes through \((-1,2,-3)\) and has direction ratios \((1,2,3)\). We test each statement.

Step 2: Statement (A):
Check whether \((-2,0,-6)\) lies on the original line. Put \(t=-1\): \((-1-1,\,2-2,\,-3-3)=(-2,0,-6)\). Yes. Direction ratios are the same \((1,2,3)\). So (A) is correct.

Step 3: Statement (B):
The plane \(x-2y+z+8=0\) has normal \((1,-2,1)\). The point \((-1,2,-3)\) gives \(-1-4-3+8=0\), so it lies on the plane. The direction dot normal is \(1-4+3=0\), so the line is parallel to the plane. Hence the line lies in the plane. (B) is correct.

Step 4: Statement (C):
For the line to be perpendicular to the plane, its direction must be parallel to the normal \((1,-2,1)\). But \((1,2,3)\) is not a multiple of \((1,-2,1)\). So (C) is incorrect.

Step 5: Statement (D):
Put \(t=1\): \((0,4,0)\). Yes, it lies on the line. (D) is correct.

Step 6: Choose:
The incorrect statement is (C).

Final Answer:
Statement C is wrong, since the direction is not parallel to the normal. \[ \boxed{C} \]