Question

The pairs of straight lines $x^2-3xy+2y^2=0$ and $x^2-3xy+2y^2+x-2=0$ form a

• A. square but not rhombus
• B. rhombus
• C. parallelogram
• D. rectangle but not a square

Hint:

If the second-degree homogeneous terms are the same in both equations, the lines are parallel, forming a parallelogram.

Answer 1

The Correct Option is C

Step 1: Separate First Pair
$x^2 - 3xy + 2y^2 = (x - 2y)(x - y) = 0$.
Lines: $L_1: x - 2y = 0$ and $L_2: x - y = 0$.
Step 2: Separate Second Pair
$x^2 - 3xy + 2y^2 + x - 2 = (x - 2y + 2)(x - y - 1) = 0$.
Lines: $L_3: x - 2y + 2 = 0$ and $L_4: x - y - 1 = 0$.
Step 3: Analyze Geometry
$L_1 \parallel L_3$ (both have coefficients 1, -2).
$L_2 \parallel L_4$ (both have coefficients 1, -1).
Two pairs of parallel lines form a parallelogram.
Step 4: Check Orthogonality
Angle between $x - 2y = 0$ and $x - y = 0$ is not 90°. Thus, it is not a rectangle.
Final Answer: (c)