Question

If the lines represented by 3x + 2py = 2 and 2x + 5y + 1 = 0 are parallel, then the value of p will be

• A. \( \frac{15}{4} \)
• B. \( -\frac{4}{5} \)
• C. \( \frac{3}{5} \)
• D. \( \frac{5}{3} \)

Hint:

Remember the conditions for different types of lines: \begin{itemize} \item Intersecting: \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \) \item Parallel: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \) \item Coincident: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \) \end{itemize} Keeping these three conditions clear will help you solve these problems quickly and accurately.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Parallel lines have the same slope but different y-intercepts. For two linear equations in standard form, this relationship can be expressed as a condition on their coefficients.
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 parallel lines, the condition is:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \] Step 3: Detailed Explanation:
First, write both equations in the standard form \( Ax + By + C = 0 \).
Equation 1: \( 3x + 2py = 2 \), which is \( 3x + 2py - 2 = 0 \).
Here, \( a_1 = 3, b_1 = 2p, c_1 = -2 \).
Equation 2: \( 2x + 5y + 1 = 0 \).
Here, \( a_2 = 2, b_2 = 5, c_2 = 1 \).
Now, apply the condition for parallel lines:
\[ \frac{a_1}{a_2} = \frac{b_1}{b_2} \] \[ \frac{3}{2} = \frac{2p}{5} \] To solve for `p`, cross-multiply:
\[ 3 \times 5 = 2 \times 2p \] \[ 15 = 4p \] \[ p = \frac{15}{4} \] We should also check the second part of the condition: \( \frac{a_1}{a_2} \neq \frac{c_1}{c_2} \).
\( \frac{3}{2} \neq \frac{-2}{1} \). This is true.
Step 4: Final Answer:
The value of `p` is \( \frac{15}{4} \).