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)

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)