Finding Value of k for Coincident Lines

Video Explanation

Question

Find the value of \(k\) for which the equations \(3x – y + 8 = 0\) and \(6x – ky + 16 = 0\) represent coincident lines.

Solution

Step 1: Identify Coefficients

\(a_1 = 3,\; b_1 = -1,\; c_1 = 8\)

\(a_2 = 6,\; b_2 = -k,\; c_2 = 16\)

Step 2: Apply Condition for Coincident Lines

For coincident lines:

\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \]

\[ \frac{3}{6} = \frac{-1}{-k} = \frac{8}{16} \]

\[ \frac{1}{2} = \frac{1}{k} = \frac{1}{2} \]

Step 3: Solve

\[ \frac{1}{k} = \frac{1}{2} \Rightarrow k = 2 \]

Final Answer

\[ \text{The lines are coincident when } k = 2. \]