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: Understanding the Concept:
This problem connects the two-point form of a line in 3D with its symmetric form. The direction ratios of the line passing through two points \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\) are given by \((x_2-x_1, y_2-y_1, z_2-z_1)\). The symmetric form of the equation uses these direction ratios as the denominators.
Step 2: Key Formula or Approach:
1. Find the direction ratios (DRs) of the line joining the points \(P_1(1, 2, 3)\) and \(P_2(3, 4, k)\). 2. The DRs are \((a, b, c) = (3-1, 4-2, k-3) = (2, 2, k-3)\). 3. The given symmetric equation is \( \frac{x-3}{1} = \frac{y-4}{1} = \frac{z-k}{5} \). The denominators of this equation, \((1, 1, 5)\), are also direction ratios for the same line. 4. Direction ratios of the same line are proportional. So, we can set up a proportion between the two sets of DRs.
Step 3: Detailed Explanation:
The direction ratios calculated from the two points are \((2, 2, k-3)\).
The direction ratios given by the symmetric equation are \((1, 1, 5)\).
Since these represent the same line, their direction ratios must be proportional. This means there is a constant \(\lambda\) such that:
\[ (2, 2, k-3) = \lambda (1, 1, 5) \] Comparing the components:
x-component: \(2 = \lambda \cdot 1 \implies \lambda = 2\)
y-component: \(2 = \lambda \cdot 1 \implies \lambda = 2\)
z-component: \(k-3 = \lambda \cdot 5\)
Using the value \(\lambda=2\) that we found from the x and y components, we can solve for k:
\[ k-3 = 2 \cdot 5 \] \[ k-3 = 10 \] \[ k = 13 \] Step 4: Final Answer:
The value of k is 13. This corresponds to option (E).