Question

For the line $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}$, which one of the following is incorrect?

• A. The line lies in the plane $x-2y+z=0$.
• B. The line is same as line $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$.
• C. The line passes through $(2,3,5)$.
• D. The line is parallel to the plane $x-2y+z-6=0$.

Answer 1

Answer: (C)

The line passes through $(2,3,5)$.

The given equation of the line is
$x-1=\frac{y}{2}-1=\frac{z}{3}-1$
$\Rightarrow x=\frac{y}{2}=\frac{z}{3}$
Hence, option B is correct.

Also, $\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}=k$(say)
$\Rightarrow x=k+1,y=2k+2,z=3k+3$.
So, any point on the line is of the form $x=k+1,y=2k+2,z=3k+3$
So, the point $(2,3,5)$ does not lie on the line.
Hence, option C is incorrect.
Also, the given line is parallel to the plane $x-2y+z-6=0$ as $1\left(1\right)+2(-2)+3\left(1\right)=0$
Hence, option D is correct.
Now, for option A, $x=k+1,y=2k+2,z=3k+3$ , we will substitute this general point of line in the equation of plane.
$k+1-2(2k+2)+3k+3=0$
Hence, the line lies in the given plane.
Hence, option $C$ is incorrect.