List of top Mathematics Questions on Equation of a Line in Space

If the foot of the perpendicular drawn from the point \((2,0,1)\) on a line passing through \((\alpha,5,1)\) is \(\left(\frac{7}{3}, \frac{5}{3}, \frac{11}{4}\right)\), then \(\alpha\) is equal to _ _ _ _ _ _
If the line through the points A(k,1,-1) and B(2k,0,2) is perpendicular to the line through the points B and C(2+2k,k,1), then the value of k is
The point $(2, 1)$ is translated parallel to the line $L : x-y = 4$ by $2\sqrt{3}$ units. If the new point $Q$ lies in the third quadrant, then the equation of the line passing through $Q$ and perpendicular to $L$ is :
The image of the line $\frac{x-1}{3}=\frac{y-3}{1}=\frac{z-4}{-5}$ in the plane $2 x - y + z+ 3 = 0$ is the line.
The coordinates of the point where the line through the points \( A(3, 4, 1) \) and \( B(5, 1, 6) \) crosses the \( XY \)-plane are