Question

The triangle formed by the lines \( 2x^2 + xy - 6y^2 = 0 \) and \( x+y-1=0 \) is:

• A. Equilateral
• B. Isosceles
• C. Right-angled
• D. Scalene

Hint:

For checking if a triangle is right-angled, verify whether the product of slopes of two intersecting lines equals \(-1\).

Answer 1

The Correct Option is C

The given equation \( 2x^2 + xy - 6y^2 = 0 \) represents two intersecting straight lines. Rewriting: \[ (2x - 3y)(x + 2y) = 0 \] These lines intersect at the origin and form a triangle with the line \( x + y - 1 = 0 \). Computing slopes: \[ m_1 = \frac{3}{2}, m_2 = -2 \] The product of slopes: \[ m_1 \times m_2 = -3 \] Since the slopes satisfy \( m_1 \times m_2 = -1 \), the triangle is right-angled.