Question

The joint equation of the lines through the origin, trisecting angles in first and third quadrants is

• A. $\sqrt{3}(x^2 - y^2) - 4xy = 0$
• B. $\sqrt{3}(x^2 + y^2) - 4xy = 0$
• C. $\sqrt{3}(x^2 + y^2) + 4xy = 0$
• D. $\sqrt{3}(x^2 - y^2) + 4xy = 0$

Answer 1

Answer: (B)

Option (B): $\sqrt{3}(x^2 + y^2) - 4xy = 0$

Answer 2

The lines making angles of 30° and 60° with the x-axis have equations

$$y=\frac{x}{\sqrt{3}} \quad \text{and} \quad y=\sqrt{3}\,x.$$

Their combined equation is

$$\left(x-\sqrt{3}\,y\right)\left(\sqrt{3}\,x-y\right)=0.$$

Expanding,

$$\sqrt{3}\,x^2-4xy+\sqrt{3}\,y^2=0,$$

which can be written as

$$\sqrt{3}(x^2+y^2)-4xy=0.$$