Final answer:
The student is seeking the equation of a line passing through (1,2,3) and intersecting another line in a three-dimensional space. Equations are developed using the direction vectors of the given line and plane, leading to the conclusion that none of the provided options are correct and the correct answer is option D 'None of these'.
Step-by-step explanation:
The student is asking for the equation of a straight line that passes through a given point and is parallel to a given plane. To find this line, first, we need the direction vector of the plane which can be represented by the coefficients of the plane equation x + 5y + 4z = 0, i.e., (1,5,4). Since the line is parallel to the plane, it will be perpendicular to this direction vector. Now, we want the line to intersect another line given by the equation x + 1 = 2(y - 2) = z + 4. The direction vector for this line can be determined by equating the components with a parameter λ and solving for x, y, and z; we get the direction vector (1,2,1).
Since the desired line has to pass through the point (1,2,3) and intersect the given line, it must share the direction vector of the line it intersects. Therefore, the direction vector for the line we are looking for is (1,2,1). Now, we can write the parametric equations of the line as:
x = 1 + t
y = 2 + 2t
z = 3 + t
However, the answer choices are given in a different format, as symmetric equations of the line. To convert these parametric equations into the symmetric form, we can express t in terms of x, y, and z, and we have:
x - 1 = t
y - 2 = 2t
z - 3 = t
Dividing the last two equations by the coefficients of t gives us the final equation:
(x - 1) / 1 = (y - 2) / 2 = (z - 3) / 1
This equation represents a line passing through the point (1,2,3) with a direction vector (1,2,1), which falls into none of the provided answer choices, indicating that the correct choice is Option D: None of these.