Let's break down how to find the equation of the line $l_2$.

Understanding Perpendicular Lines and Intersection Point

We are given two lines, $l_1$ and $l_2$, that are perpendicular to each other. This means they meet at a right angle (90 degrees). We also know the exact point where they cross, which is $(3, 4)$. This point lies on both lines.

Line Parallel to Y-Axis

The problem states that one of the lines (let's assume it's $l_1$) is parallel to the y-axis. A line parallel to the y-axis is always a vertical line. The standard equation for a vertical line is $x = c$, where $c$ is a constant representing the x-coordinate of every point on that line.

Since this line $l_1$ passes through the intersection point $(3, 4)$, its x-coordinate must be 3. Therefore, the equation for line $l_1$ is:

$x = 3$

Equation of Perpendicular Line $l_2$

Now we need to find the equation for line $l_2$. We know two things about $l_2$:

• It is perpendicular to $l_1$ (which is the vertical line $x = 3$).
• It passes through the point $(3, 4)$.

A fundamental property in coordinate geometry is that a line perpendicular to a vertical line (like $x = c$) must be a horizontal line. The standard equation for a horizontal line is $y = k$, where $k$ is a constant representing the y-coordinate of every point on that line.

Since line $l_2$ passes through the point $(3, 4)$, the y-coordinate of this point must satisfy the equation of $l_2$. Plugging the coordinates of the intersection point into the general equation for a horizontal line:

$y = k$

Substitute $y = 4$:

$4 = k$

So, the constant $k$ is 4. This gives us the specific equation for line $l_2$.

The equation of line $l_2$ is:

$y = 4$

This matches option 1.