Consider the set of all lines px+qy+r=0 such that 3p+2q+4r=0. Which one of the following statements is true ?

Options

A. The lines are all parallel.
B. Each line passes through the origin.
C. The lines are not concurrent.
D. The lines are concurrent at the point `((3)/(4),(1)/(2))`.

Answer

To solve the problem, we need to analyze the given equations and determine the relationship between the lines defined by \( px + qy + r = 0 \) and the condition \( 3p + 2q + 4r = 0 \).

### Step-by-Step Solution:

1. **Understanding the Equations**:
We have the equation of a line given by:
\[ px + qy + r = 0 \]
and a condition on the coefficients:
\[ 3p + 2q + 4r = 0 \]

2. **Expressing \( r \)**:
From the condition \( 3p + 2q + 4r = 0 \), we can express \( r \) in terms of \( p \) and \( q \):
\[ 4r = -3p - 2q \implies r = -\frac{3}{4}p - \frac{1}{2}q \]

3. **Substituting \( r \) into the Line Equation**:
Substitute \( r \) back into the line equation:
\[ px + qy - \left(\frac{3}{4}p + \frac{1}{2}q\right) = 0 \]
This simplifies to:
\[ px + qy - \frac{3}{4}p - \frac{1}{2}q = 0 \]
Rearranging gives:
\[ px + qy = \frac{3}{4}p + \frac{1}{2}q \]

4. **Finding the Points of Intersection**:
To find the points where these lines intersect, we can set the coefficients of \( p \) and \( q \) to zero:
\[ 4x - 3 = 0 \quad \text{and} \quad 2y - 1 = 0 \]

5. **Solving for \( x \) and \( y \)**:
From \( 4x - 3 = 0 \):
\[ x = \frac{3}{4} \]
From \( 2y - 1 = 0 \):
\[ y = \frac{1}{2} \]

6. **Conclusion**:
The lines \( px + qy + r = 0 \) defined by the condition \( 3p + 2q + 4r = 0 \) are concurrent at the point:
\[ \left( \frac{3}{4}, \frac{1}{2} \right) \]
Thus, the correct statement is that the lines are concurrent at the point \( \left( \frac{3}{4}, \frac{1}{2} \right) \).

### Final Answer:
The lines are concurrent at the point \( \left( \frac{3}{4}, \frac{1}{2} \right) \).

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