Question

Rhombus \( ABCD \) is given with vertices \( A(1,2) \), \( C(-3,-6) \) and sides \( AD \) and \( BC \) are parallel to the line \( 7x - y = 14 \). If coordinates of \( B \) and \( D \) are \( (\alpha,\beta) \) and \( (\gamma,\delta) \) respectively, then find \( \alpha + \beta + \gamma + \delta \):

• A. \( -4 \)
• B. \( 5 \)
• C. \( -6 \)
• D. \( -7 \)

Hint:

For quadrilaterals like rhombus, always use diagonal properties to simplify coordinate geometry problems.

Answer 1

The Correct Option is C

Step 1: Use properties of a rhombus.
In a rhombus, diagonals bisect each other at right angles. Hence, midpoint of diagonal \( AC \) is also the midpoint of diagonal \( BD \).
Step 2: Find the midpoint of \( AC \).
Coordinates of midpoint: \[ \left(\frac{1+(-3)}{2}, \frac{2+(-6)}{2}\right) = (-1,-2) \]
Step 3: Direction of sides \( AD \) and \( BC \).
Given line \( 7x - y = 14 \Rightarrow y = 7x - 14 \). So slope \( m = 7 \).
Step 4: Find coordinates of \( B \) and \( D \).
Since \( BD \) is perpendicular to \( AD \), slope of \( BD = -\frac{1}{7} \). Using midpoint \( (-1,-2) \) and slope \( -\frac{1}{7} \), coordinates of \( B \) and \( D \) are obtained.
Step 5: Compute the required sum.
Adding coordinates of \( B(\alpha,\beta) \) and \( D(\gamma,\delta) \), we get \[ \alpha + \beta + \gamma + \delta = -6 \]