Given that the lines $L_1$ and $L_2$ meet at the point $(0,4)$ and are perpendicular to each other, we can find possible equations for these lines.

Step 1: Determine the slopes

• The lines are perpendicular, meaning their slopes multiply to $-1$.
• Let the slope of $L_1$ be $m_1$, and the slope of $L_2$ be $m_2$.
• Therefore, $m_1 \times m_2 = -1$.

Step 2: Choose a slope for $L_1$

• Let’s assume the slope of $L_1$ is $m_1 = 2$.

Step 3: Calculate the slope of $L_2$

• Since $m_1 \times m_2 = -1$, the slope of $L_2$ will be $m_2 = -\frac{1}{2}$.

Step 4: Write the equations of $L_1$ and $L_2$

• The point-slope form of a line's equation is given by: $y - y_1 = m(x - x_1)$ where $(x_1, y_1)$ is a point on the line.

• For $L_1$, passing through $(0,4)$: $y - 4 = 2(x - 0)$ Simplifying, the equation of $L_1$ is: $y = 2x + 4$

• For $L_2$, passing through $(0,4)$: $y - 4 = -\frac{1}{2}(x - 0)$ Simplifying, the equation of $L_2$ is: $y = -\frac{1}{2}x + 4$

Final Answer

The possible equations for the lines $L_1$ and $L_2$ are:

• $L_1: y = 2x + 4$
• $L_2: y = -\frac{1}{2}x + 4$

Would you like any more details or have any questions?

Here are 5 related questions you might find interesting:

1. How can you determine if two lines are perpendicular just by looking at their equations?
2. What is the relationship between the slopes of parallel lines?
3. How would the equations change if the lines met at a different point, say (1, 3)?
4. Can the slope of a line ever be undefined? What does that mean geometrically?
5. How can you find the equation of a line given two points on the line?

Tip: Remember, when two lines are perpendicular, their slopes' product is always $-1$.