To solve this problem, we need to find the ratio in which the [tex]$y$[/tex]-axis divides the line segment joining the points [tex]\((-2,4)\)[/tex] and [tex]\((5,4)\)[/tex]. Additionally, we need to find the value of [tex]\( y \)[/tex] at the point of division.
### Step 1: Understand the problem
The [tex]$y$[/tex]-axis divides the line segment at some point [tex]\((0,y)\)[/tex], because on the [tex]$y$[/tex]-axis, the [tex]$x$[/tex] coordinate is always [tex]\(0\)[/tex].
Given points:
[tex]\[
A(-2, 4) \quad \text{and} \quad B(5, 4)
\][/tex]
### Step 2: Use the Section Formula
The section formula states that if a line divides a line segment joining two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of the point of division are given by:
[tex]\[
\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)
\][/tex]
In our case, the [tex]$y$[/tex]-axis divides the line segment at [tex]\(( 0, y )\)[/tex].
For the [tex]$x$[/tex] coordinate:
[tex]\[
\frac{m \cdot x_2 + n \cdot x_1}{m + n} = 0
\][/tex]
### Step 3: Substitute Known Values
Let's substitute [tex]\((x_1, y_1) = (-2, 4)\)[/tex] and [tex]\((x_2, y_2) = (5, 4)\)[/tex]:
[tex]\[
\frac{m \cdot 5 + n \cdot (-2)}{m + n} = 0
\][/tex]
### Step 4: Solve for the Ratio [tex]\( m:n \)[/tex]
[tex]\[
5m - 2n = 0
\][/tex]
Solve for [tex]\( m \)[/tex] in terms of [tex]\( n \)[/tex]:
[tex]\[
5m = 2n \implies \frac{m}{n} = \frac{2}{5} \implies m:n = 2:5
\][/tex]
### Step 5: Find the [tex]\( y \)[/tex]-coordinate at Point of Division
Since both points have the same [tex]\( y \)[/tex]-coordinate (4), the [tex]\( y \)[/tex]-coordinate of the division point on the [tex]$y$[/tex]-axis will also be [tex]\(4\)[/tex].
### Conclusion
The line segment joining the points [tex]\((-2, 4)\)[/tex] and [tex]\((5, 4)\)[/tex] is divided by the [tex]$y$[/tex]-axis in the ratio [tex]\(2:5\)[/tex], and the [tex]\( y \)[/tex]-coordinate at the point of division is [tex]\(4\)[/tex].
Thus, the ratio is [tex]\(2:5\)[/tex] and the [tex]\( y \)[/tex]-coordinate is [tex]\(4\)[/tex].
