To find the ratio in which the point [tex]\((2, y)\)[/tex] divides the line segment joining [tex]\((-4, 3)\)[/tex] and [tex]\((6, 3)\)[/tex], as well as the value of [tex]\(y\)[/tex], we can use the section formula.
Let's denote the points as follows:
- [tex]\(A(x_1, y_1) = (-4, 3)\)[/tex]
- [tex]\(B(x_2, y_2) = (6, 3)\)[/tex]
- [tex]\(P(x, y) = (2, y)\)[/tex]
Since the y-coordinates of both [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are the same, this implies they lie on a horizontal line. Therefore, the y-coordinate of the point [tex]\(P\)[/tex] is also the same. Hence,
[tex]\[ y = 3 \][/tex]
Now, let's find the ratio in which the point [tex]\(P\)[/tex] divides the line segment [tex]\(AB\)[/tex]. Let the ratio be [tex]\(m:n\)[/tex].
We use the section formula for the x-coordinates:
[tex]\[ x = \frac{m \cdot x_2 + n \cdot x_1}{m + n} \][/tex]
Substitute the given values:
[tex]\[ 2 = \frac{m \cdot 6 + n \cdot (-4)}{m + n} \][/tex]
[tex]\[ 2 = \frac{6m - 4n}{m + n} \][/tex]
Cross-multiply to clear the fraction:
[tex]\[ 2(m + n) = 6m - 4n \][/tex]
[tex]\[ 2m + 2n = 6m - 4n \][/tex]
Rearrange the equation to isolate terms involving [tex]\(m\)[/tex] and [tex]\(n\)[/tex]:
[tex]\[ 2m + 2n + 4n = 6m \][/tex]
[tex]\[ 2m + 6n = 6m \][/tex]
Subtract [tex]\(2m\)[/tex] from both sides:
[tex]\[ 6n = 4m \][/tex]
Divide both sides by 2:
[tex]\[ 3n = 2m \][/tex]
To find the ratio [tex]\( m : n \)[/tex], divide both sides by [tex]\( n \)[/tex]:
[tex]\[ \frac{3n}{n} = \frac{2m}{n} \][/tex]
[tex]\[ 3 = \frac{2m}{n} \][/tex]
Thus,
[tex]\[ \frac{m}{n} = \frac{3}{2} \][/tex]
Therefore, the ratio [tex]\( m : n \)[/tex] is [tex]\( 3 : 2 \)[/tex].
In summary:
1. The point [tex]\((2, y)\)[/tex] divides the line segment joining [tex]\((-4, 3)\)[/tex] and [tex]\((6, 3)\)[/tex] in the ratio [tex]\(3:2\)[/tex].
2. The value of [tex]\(y\)[/tex] is [tex]\(3\)[/tex].
