The diagonals of parallelogram PQRS are along the lines `x+3y=4` and `6x-2y=7`. Then PQRS must be a

Options

A. rectangle
B. square
C. cyclic quadrilateral
D. rhombus

Answer

To determine the type of parallelogram PQRS given that its diagonals are along the lines \( x + 3y = 4 \) and \( 6x - 2y = 7 \), we will follow these steps:

### Step 1: Identify the equations of the diagonals
The equations of the diagonals are:
1. \( L_1: x + 3y = 4 \)
2. \( L_2: 6x - 2y = 7 \)

### Step 2: Convert the equations to slope-intercept form
To find the slopes of these lines, we will convert them into the slope-intercept form \( y = mx + b \).

For \( L_1: x + 3y = 4 \):
\[ 3y = -x + 4 \implies y = -\frac{1}{3}x + \frac{4}{3} \]
Thus, the slope \( m_1 = -\frac{1}{3} \).

For \( L_2: 6x - 2y = 7 \):
\[ -2y = -6x + 7 \implies 2y = 6x - 7 \implies y = 3x - \frac{7}{2} \]
Thus, the slope \( m_2 = 3 \).

### Step 3: Check if the diagonals are perpendicular
To check if the diagonals are perpendicular, we multiply their slopes:
\[ m_1 \cdot m_2 = \left(-\frac{1}{3}\right) \cdot 3 = -1 \]
Since the product of the slopes is \(-1\), the lines are indeed perpendicular.

### Step 4: Conclude the type of parallelogram
In a parallelogram, if the diagonals are perpendicular to each other, then the parallelogram is a rhombus. Therefore, we conclude that the parallelogram PQRS must be a rhombus.

### Final Answer
The given parallelogram PQRS must be a rhombus.

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