3D Geometry
Vector 3D
nta_abhyas_2025
Grade None
Question:
The measure made by the $x, y$ and $z$ axis, by the plane which bisects the line joining the points $(1, 2, 3)$ and $(-3, 4, 5)$ at right angles, are $\alpha$, $\beta$ and $\gamma$ respectively, then the ordered triplet $(\alpha, \beta, \gamma)$ is
-2, 3, 0
2, 9, 0
0, -2/3, 0
0, 1/3, 0
Step-by-Step Solution
Key Concept: A line is parallel to the normal of a plane when its direction vector is proportional to the plane's normal vector.
The equation of the plane is $\frac{x}{4} + \frac{y}{6} + \frac{z}{1} = 1$. The normal to the plane is $(\frac{1}{4}, \frac{1}{6}, 1)$, or equivalently $(3, 2, 12)$ after scaling. The midpoint $M$ of line joining $P(1, 2, 3)$ and $Q(-3, 4, 5)$ is $M = (-1, 3, 4)$. Since $PQ$ is parallel to the normal, we have $PQ = \lambda(3, 2, 12)$. For a point on the plane, checking $\frac{x}{4} + \frac{y}{6} + \frac{z}{1} = 1$ and solving using the parallel condition yields the ordered pair $(\frac{7}{2}, 9, 9)$.
Correct Answer: 1