Question

If a point \( P \) with \( x \)-coordinate \( 7 \) lies on the line joining the points \( A(1,2,3) \) and \( B(4,6,8) \), then the coordinates of the point \( P \) are

• A. \( (7,10,-13) \)
• B. \( (7,-10,13) \)
• C. \( (7,10,12) \)
• D. \( (7,10,13) \)
• E. \( (7,10,15) \)

Hint:

To find a point on a line joining two points, first write the parametric form using the direction vector, then use the given coordinate condition to determine the parameter.

Answer 1

The Correct Option is D

Step 1: Find the direction vector of the line joining \( A \) and \( B \).
Given \[ A=(1,2,3), \qquad B=(4,6,8) \] the direction vector of the line \( AB \) is \[ \overrightarrow{AB}=B-A=(4-1,\ 6-2,\ 8-3) \] \[ \overrightarrow{AB}=(3,4,5) \]

Step 2: Write the parametric equation of the line through \( A \).
A general point on the line through \( A \) in the direction \( \overrightarrow{AB} \) is \[ (x,y,z)=(1,2,3)+t(3,4,5) \] So, \[ x=1+3t,\qquad y=2+4t,\qquad z=3+5t \]

Step 3: Use the given condition \( x=7 \).
It is given that the \( x \)-coordinate of point \( P \) is \( 7 \).
So from \[ x=1+3t \] we get \[ 7=1+3t \] \[ 3t=6 \] \[ t=2 \]

Step 4: Find the corresponding \( y \)-coordinate.
Using \( t=2 \) in \[ y=2+4t \] we get \[ y=2+4(2)=2+8=10 \]

Step 5: Find the corresponding \( z \)-coordinate.
Using \( t=2 \) in \[ z=3+5t \] we get \[ z=3+5(2)=3+10=13 \]

Step 6: Write the coordinates of \( P \).
Thus the required point is \[ P=(7,10,13) \] Let us verify quickly that this point lies on the same line: \[ \frac{7-1}{3}=\frac{10-2}{4}=\frac{13-3}{5}=2 \] So the point is correct.

Step 7: Final conclusion.
Hence, the coordinates of \( P \) are \[ \boxed{(7,10,13)} \] Therefore, the correct option is \[ \boxed{(4)\ (7,10,13)} \]