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Question: For the line \[\dfrac{{{\text{x - 1}}}}{{\text{1}}}{\text{ = }}\dfrac{{{\text{y - 2}}}}{{\text{2}}}{...

For the line x - 11 = y - 22 = z - 33\dfrac{{{\text{x - 1}}}}{{\text{1}}}{\text{ = }}\dfrac{{{\text{y - 2}}}}{{\text{2}}}{\text{ = }}\dfrac{{{\text{z - 3}}}}{{\text{3}}}, which one of the following is incorrect ?
A) The line lies on the plane x - 2y + z = 0{\text{x - 2y + z = 0}}
B) The line is same as x1 = y2 = z3\dfrac{{\text{x}}}{{\text{1}}}{\text{ = }}\dfrac{{\text{y}}}{{\text{2}}}{\text{ = }}\dfrac{{\text{z}}}{{\text{3}}}
C) The line passes through (2,3,5){\text{(2,3,5)}}
D) The line is parallel to the plane x - 2y + z - 6 = 0{\text{x - 2y + z - 6 = 0}}

Explanation

Solution

First of all, split the given equation of the line and also calculate it’s parametric points so that we can satisfy those points in given equation of plane and verify it. Hence, proceed for the solution by checking out each and every option individually.

Complete step by step solution: Given equation of line x - 11 = y - 22 = z - 33\dfrac{{{\text{x - 1}}}}{{\text{1}}}{\text{ = }}\dfrac{{{\text{y - 2}}}}{{\text{2}}}{\text{ = }}\dfrac{{{\text{z - 3}}}}{{\text{3}}}
Let the given equation of line be represented as x - 11 = y - 22 = z - 33 = k\dfrac{{{\text{x - 1}}}}{{\text{1}}}{\text{ = }}\dfrac{{{\text{y - 2}}}}{{\text{2}}}{\text{ = }}\dfrac{{{\text{z - 3}}}}{{\text{3}}}{\text{ = k}}
Now, simplifying the points (x, y, z)  = (k + 1,2k + 2,3k + 3){\text{ = (k + 1,2k + 2,3k + 3)}}
Now, substitute the value of point in the given equation of plane, for option (a) , we get,

x - 2y + z = 0 ⇒k + 1 - 2(2k + 2) + 3k + 3 = 0 ⇒0 = 0  {\text{x - 2y + z = 0}} \\\ \Rightarrow {\text{k + 1 - 2(2k + 2) + 3k + 3 = 0}} \\\ \Rightarrow {\text{0 = 0}} \\\

Hence option (a) is correct.
For option (b) , rearranging the equation of given line
x1 - 1 = y2 - 1 = z3 - 1\dfrac{{\text{x}}}{{\text{1}}}{\text{ - 1 = }}\dfrac{{\text{y}}}{{\text{2}}}{\text{ - 1 = }}\dfrac{{\text{z}}}{{\text{3}}}{\text{ - 1}}
Hence the equation can be presented as , x1 = y2 = z3\dfrac{{\text{x}}}{{\text{1}}}{\text{ = }}\dfrac{{\text{y}}}{{\text{2}}}{\text{ = }}\dfrac{{\text{z}}}{{\text{3}}}
So option (b) is also correct.
Now, for option (c) as ,
(2,3,5)≠(k + 1,2k + 2,3k + 3){\text{(2,3,5)}} \ne {\text{(k + 1,2k + 2,3k + 3)}}for any value of k
So option (c) is incorrect .

So, option (c) is our required answer.

Note: A line is a collection of points in space which satisfy an equation. A (geometric) vector can be thought of as "a direction and a magnitude", and can be represented by a single point in space.