To find the joint equation of lines passing through the origin that make angles of 30° and 60° with the positive x-axis, we can use the general equation for a straight line in the plane:

$y = mx$

where $m$ is the slope of the line. The slope $m$ of a line making an angle $\theta$ with the positive x-axis is given by:

$m = \tan(\theta)$

Steps:

1. For the line making an angle of 30° with the positive x-axis:

   • The slope $m_1 = \tan(30^\circ)$.
   • We know $\tan(30^\circ) = \frac{1}{\sqrt{3}}$.
   • So, the equation of the line is: $y = \frac{1}{\sqrt{3}}x$

2. For the line making an angle of 60° with the positive x-axis:

   • The slope $m_2 = \tan(60^\circ)$.
   • We know $\tan(60^\circ) = \sqrt{3}$.
   • So, the equation of the line is: $y = \sqrt{3}x$

Joint Equation of the Two Lines:

To find the joint equation of these two lines, we multiply their individual equations:

$\left( y - \frac{1}{\sqrt{3}}x \right) \left( y - \sqrt{3}x \right) = 0$

Expanding this product:

$y^2 - \left(\frac{1}{\sqrt{3}}x + \sqrt{3}x\right)y + \left(\frac{1}{\sqrt{3}}x \cdot \sqrt{3}x\right) = 0$

Simplifying:

$y^2 - \left(\frac{1}{\sqrt{3}} + \sqrt{3}\right)xy + x^2 = 0$

To further simplify $\frac{1}{\sqrt{3}} + \sqrt{3}$:

$\frac{1}{\sqrt{3}} + \sqrt{3} = \frac{1 + 3}{\sqrt{3}} = \frac{4}{\sqrt{3}}$

So the joint equation becomes:

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

This is the joint equation of the two lines passing through the origin with angles of 30° and 60° with the x-axis.

Would you like more details or have any questions?

Relative Questions:

1. What would the joint equation be if the angles were 45° and 135°?
2. How does the joint equation change if the lines do not pass through the origin?
3. Can you derive the joint equation for lines making complementary angles?
4. How do the slopes of the lines affect the nature of the joint equation?
5. What is the geometric interpretation of the joint equation of two lines?

Tip:

When multiplying linear equations to find a joint equation, always simplify the intermediate steps to ensure accuracy and ease of solving.