The combine equation of the lines passing through the origin and each of which makes an angle of `60^(@)` with the Y-axis is

Options

A. `x^(2)-3y^(2)=0`
B. `3x^(2)-y^(2)=0`
C. `x^(2)-sqrt(3)y^(2)=0`
D. `sqrt(3)x^(2)-y^(2)=0`

Answer

To find the combined equation of the lines passing through the origin and making an angle of \(60^\circ\) with the Y-axis, we can follow these steps:

### Step 1: Determine the angles with respect to the X-axis
Since the lines make an angle of \(60^\circ\) with the Y-axis, we can find the angles with respect to the X-axis. The angles with respect to the X-axis will be:
- \( \theta_1 = 90^\circ - 60^\circ = 30^\circ \)
- \( \theta_2 = 90^\circ + 60^\circ = 150^\circ \)

### Step 2: Find the slopes of the lines
The slope \(m\) of a line that makes an angle \(\theta\) with the X-axis is given by:
\[ m = \tan(\theta) \]

Calculating the slopes for both angles:
- For \( \theta_1 = 30^\circ \):
\[ m_1 = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]
- For \( \theta_2 = 150^\circ \):
\[ m_2 = \tan(150^\circ) = -\frac{1}{\sqrt{3}} \]

### Step 3: Write the equations of the lines
The general equation of a line passing through the origin with slope \(m\) is given by:
\[ y = mx \]

Thus, the equations of the lines are:
- For \(m_1 = \frac{1}{\sqrt{3}}\):
\[ y = \frac{1}{\sqrt{3}}x \]
- For \(m_2 = -\frac{1}{\sqrt{3}}\):
\[ y = -\frac{1}{\sqrt{3}}x \]

### Step 4: Combine the equations
To find the combined equation of the two lines, we can express both equations in a standard form:
1. From \(y = \frac{1}{\sqrt{3}}x\), we can rearrange it to:
\[ \sqrt{3}y - x = 0 \]
2. From \(y = -\frac{1}{\sqrt{3}}x\), we can rearrange it to:
\[ \sqrt{3}y + x = 0 \]

### Step 5: Form the combined equation
The combined equation of the lines can be obtained by multiplying the two linear equations:
\[ (\sqrt{3}y - x)(\sqrt{3}y + x) = 0 \]

Expanding this gives:
\[ 3y^2 - x^2 = 0 \]

### Final Answer
The combined equation of the lines passing through the origin and making an angle of \(60^\circ\) with the Y-axis is:
\[ 3y^2 - x^2 = 0 \]

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