Question

If the point \( (3,6,k) \) lies on the line \( \dfrac{x-1}{1}=\dfrac{y-2}{2}=\dfrac{z-3}{3} \), then the value of \( k \) is

• A. \( 2 \)
• B. \( 3 \)
• C. \( 9 \)
• D. \( -2 \)
• E. \( -3 \)

Hint:

For a point lying on a line given in symmetric form, first set the common ratio equal to a parameter and convert the line into parametric equations. Then compare coordinates one by one.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
If a point lies on a line, its coordinates must satisfy the equation of the line. We can substitute the given coordinates into the symmetric equation of the line to find the unknown value k.
Step 2: Key Formula or Approach:
Substitute the coordinates of the point \((x, y, z) = (3, 6, k)\) into the line equation \( \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3} \). All three parts of the equation must be equal.
Step 3: Detailed Explanation:
The given line equation is:
\[ \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3} \] The given point is (3, 6, k). Let's substitute x=3, y=6, and z=k into the equation.
\[ \frac{3-1}{1} = \frac{6-2}{2} = \frac{k-3}{3} \] Now, let's evaluate the first two parts to find the common ratio:
\[ \frac{2}{1} = \frac{4}{2} \] \[ 2 = 2 \] This confirms that the x and y coordinates are consistent with the point being on the line. Now we use this common ratio to solve for k:
\[ \frac{k-3}{3} = 2 \] Multiply both sides by 3:
\[ k-3 = 6 \] \[ k = 9 \] Step 4: Final Answer:
The value of k is 9. This corresponds to option (C).