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: Equation of Line
Direction ratios of the line are $(2-1, 2-1, 2-1) = (1, 1, 1)$.
Equation: $\frac{x-1}{1} = \frac{y-1}{1} = \frac{z-1}{1} = \lambda$.
General point on line: $(1+\lambda, 1+\lambda, 1+\lambda)$.

Step 2: Solve for $\lambda$
Substitute this point into the plane $x + y + z = 9$:
$(1+\lambda) + (1+\lambda) + (1+\lambda) = 9 \implies 3 + 3\lambda = 9 \implies 3\lambda = 6 \implies \lambda = 2$.

Step 3: Find Coordinates
Substitute $\lambda = 2$ back into the general point:
$x = 1+2 = 3$, $y = 1+2 = 3$, $z = 1+2 = 3$.

Step 4: Conclusion
The point of intersection is $(3, 3, 3)$.
Final Answer: (D)