Question

The triangle formed by \( x^2 - 4xy + y^2 = 0 \) and \( x + y + 4\sqrt{6} = 0 \) is:

• A. An equilateral triangle
• B. A right-angled triangle
• C. An isosceles triangle
• D. A scalene triangle

Hint:

Triangle from Pair of Lines}
Homogeneous quadratic in \( x, y \) often means pair of straight lines
Use angle between lines and intersecting line to check triangle type
An equilateral triangle has all internal angles as 60°

Answer 1

The Correct Option is A

Given: \[ x^2 - 4xy + y^2 = 0 \Rightarrow (x - y)^2 = 4xy \Rightarrow \text{Pair of straight lines} \] This is a homogeneous equation, representing two lines intersecting at the origin. Let us factor: \[ x^2 - 4xy + y^2 = 0 \Rightarrow \text{Lines: } x = ky \text{ where } k = \tan\theta \Rightarrow \theta = \text{angle between lines} \] Using angle between lines formula: \[ \tan\theta = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|, \text{ and angle between } x - 2y = 0 \text{ and } x - y = 0 \text{ is } 60^\circ \] The third line \( x + y + 4\sqrt{6} = 0 \) intersects these two at equal angles forming 60° each Hence, the triangle has three 60° angles ⇒ equilateral.