Question

The joint equation of two lines through the origin, each of which makes an angle of \(30^\circ\) with the line \(x + y = 0\), is

• A. \(x^2 + 4xy - y^2 = 0\)
• B. \(x^2 - 4xy + y^2 = 0\)
• C. \(x^2 + 4xy + y^2 = 0\)
• D. \(x^2 - 4xy - y^2 = 0\)

Hint:

For joint equations of lines through the origin, always form the combined quadratic equation using the slopes.

Answer 1

The Correct Option is C

Step 1: Find the slope of the given line.
The line \(x + y = 0\) can be written as \(y = -x\). Hence, the slope is \(m_1 = -1\).

Step 2: Use the angle formula between two lines.
If a line with slope \(m\) makes an angle \(\theta\) with another line of slope \(m_1\), then \[ \tan \theta = \left| \frac{m - m_1}{1 + m m_1} \right| \]
Step 3: Substitute \(\theta = 30^\circ\).
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} = \left| \frac{m + 1}{1 - m} \right| \]
Step 4: Solve for slopes and form the joint equation.
Solving gives two slopes, whose combined equation through origin is \[ (x + y)^2 + 2xy = 0 \Rightarrow x^2 + 4xy + y^2 = 0 \]