Question

The equation of the straight line joining the points \((1,2,3)\) and \((3,4,k)\) is \(\frac{x-3}{1}=\frac{y-4}{1}=\frac{z-k}{5}\). Then the value of \(k\) is

• A. \(4\)
• B. \(5\)
• C. \(7\)
• D. \(10\)
• E. \(13\)

Hint:

For 3D lines, direction ratios from two points must match the direction ratios from the given line equation. Always compare ratios carefully.

Answer 1

Answer: (E)

Step 1: Identify the given points.
The two points are:
\[ A(1,2,3), \quad B(3,4,k) \]

Step 2: Find the direction ratios from the two points.
Direction ratios of the line joining \(A\) and \(B\) are:
\[ (3-1,\ 4-2,\ k-3)=(2,\ 2,\ k-3) \]

Step 3: Identify direction ratios from the given line.
From the symmetric equation:
\[ \frac{x-3}{1}=\frac{y-4}{1}=\frac{z-k}{5} \] the direction ratios are:
\[ (1,\ 1,\ 5) \]

Step 4: Use proportionality of direction ratios.
Since both represent the same line, their direction ratios must be proportional:
\[ \frac{2}{1}=\frac{2}{1}=\frac{k-3}{5} \]

Step 5: Compare ratios.
\[ \frac{2}{1}=2 \] So: \[ \frac{k-3}{5}=2 \]

Step 6: Solve for \(k\).
\[ k-3=10 \] \[ k=13 \]

Step 7: Match with the options.
Thus, the value of \(k\) is:
\[ \boxed{13} \] which matches option \((5)\).