Question

Show that the four lines given by the equations 2x² + 3xy - 2y² = 0 and 2x² + 3xy - 2y² + 3x + y + 1 = 0 form a square.

Answer 1

We know that the lines represented by the equations are

2x - y = 0

x + 2y = 0

2x - y + 1 = 0

and 2x + 2y + 1 = 0

Solving 2x - y = 0 and x + 2y + 1 = 0, we have

x =- 3/5 and y = -1/5

so that the slope of diagonal through origin is 1/3. Since the other diagonal is 3x + y + 1 = 0, the diagonals are at right angles. Therefore, the parallelogram is a square.