Question

The line passing through the points $(2,1,0) , (5,0,1)$ and $(4,1,1)$ intersects the $x-$axis at

• A. $(3,0,0)$
• B. $(-3,0,0)$
• C. $(0,0,0)$
• D. $(1,0,0)$
• E. (-1,0,0)
• F. $(5,0,0)$

Answer 1

Answer: (F)

$(5,0,0)$

Answer 2

Given that

The given points are $(2, 1, 0), (5, 0, 1)$, and $(4, 1, 1)$

Then to find where it intersects the $x-$axis, we first need to find the parametric equation of the line.

Let the parametric equation of the line be:

$r = (2, 1, 0) + t × (a, b, c)$

where $(2, 1, 0)$ is one of the given points, and $(a, b, c)$ is the direction vector of the line.

We can find the direction vector by subtracting one point from another. Let's choose $(5, 0, 1)$ and $(2, 1, 0)$:

Direction vector = $(5, 0, 1) - (2, 1, 0) = (3, -1, 1)$

So, the parametric equation of the line is:

$r = (2, 1, 0) + t × (3, -1, 1)$

Now, to find where this line intersects the x-axis, we need to find the value of t when the y and z coordinates become zero. That means we need to solve the following system of equations:

1. $x = 2 + 3t$
2. $y = 1 - t = 0$
3. $z = t = 0$

From equation (2), we find $t = 1.$

Now, substitute $t = 1$,

into equation (1) to find the $x-$coordinate:

$x = 2 + 3(1) = 5$

So, the line intersects the $x-$axis at the point $(5, 0, 0).$ (_Ans)