Question

Identify the co-ordinates of the point where the line joining \( (1,1,1) \) and \( (2,2,2) \) intersects the plane \( x+y+z=9 \).

• A. \( (2,2,2) \)
• B. \( (3,3,3) \)
• C. \( (4,4,4) \)
• D. \( (1,1,1) \)

Hint:

If a line passes through points proportional like \((1,1,1)\), \((2,2,2)\), the line direction is \((1,1,1)\), making parametric substitution very quick.

Answer 1

The Correct Option is B

Concept: The equation of a line passing through points \(A\) and \(B\) can be written using vector form: \[ (x,y,z) = (x_1,y_1,z_1) + t(x_2-x_1,\;y_2-y_1,\;z_2-z_1) \]

Step 1: Find the parametric form of the line. Points: \[ A=(1,1,1), \quad B=(2,2,2) \] Direction ratios: \[ (2-1,2-1,2-1)=(1,1,1) \] Thus, \[ (x,y,z)=(1,1,1)+t(1,1,1) \] \[ x=1+t,\quad y=1+t,\quad z=1+t \]

Step 2: Substitute into the plane equation. \[ x+y+z=9 \] \[ (1+t)+(1+t)+(1+t)=9 \] \[ 3+3t=9 \] \[ t=2 \]

Step 3: Find the intersection point. \[ x=1+2=3,\quad y=3,\quad z=3 \] Thus, the point of intersection is: \[ (3,3,3) \]