3D Geometry
Three Dimensional Geometry
star_batch_jee_advanced_2025
Grade 12
Question:
If $a, b, c$ be the lengths of the intercepts of the plane passing through the intersection of the planes $2x + y + 2z = 9, 4x - 5y - 4z = 1$ and the point $(3, 2, 1)$ on the coordinate axes, then $(5a + b + c)/2 = $ __________.
Step-by-Step Solution
Key Concept: Use the family of planes through the intersection of two given planes: $(P_1) + \lambda(P_2) = 0$, apply the point condition to find $\lambda$, then extract intercepts from the resulting plane equation.
The plane passes through the line of intersection of $2x + y + 2z = 9$ and $4x - 5y - 4z = 1$, so its equation is $(2x + y + 2z - 9) + \lambda(4x - 5y - 4z - 1) = 0$. Simplifying: $(2 + 4\lambda)x + (1 - 5\lambda)y + (2 - 4\lambda)z = 9 + \lambda$. Since it passes through $(3, 2, 1)$: $(2 + 4\lambda)(3) + (1 - 5\lambda)(2) + (2 - 4\lambda)(1) = 9 + \lambda$, which gives $6 + 12\lambda + 2 - 10\lambda + 2 - 4\lambda = 9 + \lambda$, so $10 - 2\lambda = 9 + \lambda$, yielding $\lambda = 1/3$. The plane equation becomes $ rac{10}{3}x - rac{2}{3}y + rac{2}{3}z = rac{28}{3}$, or $10x - 2y + 2z = 28$, or $5x - y + z = 14$. The intercepts are $a = 14/5$, $b = -14$, $c = 14$. Therefore $(5a + b + c)/2 = (14 - 14 + 14)/2 = 7$.
Correct Answer: 7