Question

The co-ordinates of the point in which line joining $(1, 1, 1)$ and $(2, 2, 2)$ intersects the plane $x + y + z = 9$ are

• A. (3, 4, 2)
• B. (2, 3, 4)
• C. (3, 2, 4)
• D. (3, 3, 3)

Hint:

If a line passes through the origin or has $x=y=z$ form, and the plane is $x+y+z=d$, the intersection point is always $(d/3, d/3, d/3)$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
We need to find the intersection point of a line defined by two points and a given plane.
Step 2: Detailed Explanation:
1. Equation of the line through \( (1, 1, 1) \) and \( (2, 2, 2) \):
Direction ratios are \( (2-1, 2-1, 2-1) = (1, 1, 1) \).
Line equation: \( \frac{x-1}{1} = \frac{y-1}{1} = \frac{z-1}{1} = \lambda \).
2. General point on the line: \( (1+\lambda, 1+\lambda, 1+\lambda) \).
3. This point lies on the plane \( x + y + z = 9 \).
\[ (1+\lambda) + (1+\lambda) + (1+\lambda) = 9 \]
\[ 3(1+\lambda) = 9 \implies 1+\lambda = 3 \implies \lambda = 2 \]
4. The coordinates are \( (1+2, 1+2, 1+2) = (3, 3, 3) \).
Step 4: Final Answer:
The point of intersection is \( (3, 3, 3) \).