3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
The orthogonal projection of the line $\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-1}{4}$ on the plane $3x + 4y + 5z = 0$ is:
$\frac{x}{7} = \frac{y}{1} = \frac{z}{-5}$
$\frac{x}{2} = \frac{y}{3} = \frac{z}{4}$
$\frac{x}{1} = \frac{y}{-2} = \frac{z}{1}$
None of these
Step-by-Step Solution
Key Concept: Finding the orthogonal projection requires finding a plane through the line that is orthogonal to the given plane, then finding the line of intersection.
The plane passing through the given line has equation $3x - 2y + 1 + \lambda(2(z - z + 1)) = 0$. For orthogonality with plane $3x - 2y + z = 0$, we need $3(3 + 2\lambda) + 4(-2) + 5(-\lambda) = 0$, giving $\lambda = -1$. The required plane is $3x + 4y + 5z = 0$, and the orthogonal projection of the line is $\frac{x}{7} = \frac{y}{1} = \frac{z}{-5}$.
Correct Answer: 1