Question

For which value of p, the equations 3x - y + 8 = 0 and 6x - py = 16 represents coincident lines?

• A. -2
• B. 2
• C. \( \frac{1}{2} \)
• D. \( -\frac{1}{2} \)

Hint:

In competitive exams, if you find a contradiction like \( 1/2 = -1/2 \), double-check your work. If it's still there, it's likely a typo in the question. Assume the most plausible typo that makes the problem solvable and matches one of the options. Here, changing the sign of the constant term is the most common type of error.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Coincident lines are lines that completely overlap. This means they are essentially the same line. For two linear equations to represent coincident lines, their corresponding coefficients must be in proportion.
Step 2: Key Formula or Approach:
For two linear equations \( a_1x + b_1y + c_1 = 0 \) and \( a_2x + b_2y + c_2 = 0 \) to represent coincident lines, the condition is:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \] Step 3: Detailed Explanation:
First, let's write both equations in the standard form \( Ax + By + C = 0 \).
Equation 1: \( 3x - y + 8 = 0 \)
Here, \( a_1 = 3, b_1 = -1, c_1 = 8 \).
Equation 2: \( 6x - py = 16 \), which can be rewritten as \( 6x - py - 16 = 0 \).
Here, \( a_2 = 6, b_2 = -p, c_2 = -16 \).
Now, apply the condition for coincident lines:
\[ \frac{3}{6} = \frac{-1}{-p} = \frac{8}{-16} \] Let's simplify the known ratios:
\[ \frac{1}{2} = \frac{1}{p} = -\frac{1}{2} \] This leads to a contradiction, as \( \frac{1}{2} \neq -\frac{1}{2} \). This indicates a likely typo in the original question. A common form for such problems is for the second equation to be a direct multiple of the first. Let's assume the second equation should have been `6x - py + 16 = 0` for the ratios to match.
Corrected Interpretation:
Let's assume the second equation is \( 6x - py + 16 = 0 \).
In this case, \( a_2 = 6, b_2 = -p, c_2 = 16 \).
The condition becomes:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \] \[ \frac{3}{6} = \frac{-1}{-p} = \frac{8}{16} \] Simplifying the ratios gives:
\[ \frac{1}{2} = \frac{1}{p} = \frac{1}{2} \] From the equation \( \frac{1}{2} = \frac{1}{p} \), we can solve for `p`.
\[ p = 2 \] Step 4: Final Answer:
Assuming the intended equation was `6x - py + 16 = 0`, the value of `p` is 2.