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 conditions for the lines represented by the equation \( px + qy + r = 0 \) under the constraint \( 3p + 2q + 4r = 0 \).

### Step-by-Step Solution:

1. **Write the equations**:
We have two equations:
- Equation of the line: \( px + qy + r = 0 \) (let's call this Equation 1)
- Constraint: \( 3p + 2q + 4r = 0 \) (let's call this Equation 2)

2. **Express one variable in terms of others**:
From Equation 2, we can express \( r \) in terms of \( p \) and \( q \):
\[ 4r = -3p - 2q \implies r = -\frac{3}{4}p - \frac{1}{2}q \]

3. **Substitute \( r \) back into Equation 1**:
Substitute the expression for \( r \) into Equation 1:
\[ px + qy - \left(\frac{3}{4}p + \frac{1}{2}q\right) = 0 \]
Simplifying this gives:
\[ px + qy - \frac{3}{4}p - \frac{1}{2}q = 0 \]
Rearranging terms:
\[ px - \frac{3}{4}p + qy - \frac{1}{2}q = 0 \]
Factor out \( p \) and \( q \):
\[ p\left(x - \frac{3}{4}\right) + q\left(y - \frac{1}{2}\right) = 0 \]

4. **Analyze the condition for lines**:
The above equation represents a family of lines that will pass through the point \( \left(\frac{3}{4}, \frac{1}{2}\right) \). This means that all lines of the form \( px + qy + r = 0 \) under the given constraint will intersect at this point.

5. **Conclusion**:
Thus, the lines represented by \( px + qy + r = 0 \) with the constraint \( 3p + 2q + 4r = 0 \) are concurrent at the point \( \left(\frac{3}{4}, \frac{1}{2}\right) \).

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

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