To calculate where the line [tex]\( y = 2 \)[/tex] divides the line segment joining the points [tex]\( A(6,5) \)[/tex] and [tex]\( B(4,-3) \)[/tex], we can make use of the section formula. Let's proceed through the steps needed to find the coordinates of the dividing point.
1. Identify the Coordinates and the Line Equation:
- Point [tex]\( A \)[/tex] has coordinates [tex]\( (6,5) \)[/tex].
- Point [tex]\( B \)[/tex] has coordinates [tex]\( (4,-3) \)[/tex].
- The line divides the segment at [tex]\( y = 2 \)[/tex].
2. Use the Section Formula:
The section formula helps us to find a point that divides a line segment in a given ratio. For a line segment joining points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex], if it is divided in the ratio [tex]\( m:n \)[/tex], the coordinates of the dividing point [tex]\((x, y)\)[/tex] are given by:
[tex]\[
x = \frac{m x_2 + n x_1}{m + n}
\][/tex]
[tex]\[
y = \frac{m y_2 + n y_1}{m + n}
\][/tex]
3. Set Up the Equation Based on the y-coordinate:
Since the line divides the segment at [tex]\( y = 2 \)[/tex]:
[tex]\[
2 = \frac{m y_B + n y_A}{m + n} = \frac{m (-3) + n (5)}{m + n}
\][/tex]
4. Solve for the Ratio [tex]\( \frac{m}{n} \)[/tex]:
[tex]\[
2(m + n) = m(-3) + n (5)
\][/tex]
[tex]\[
2m + 2n = -3m + 5n
\][/tex]
[tex]\[
5m = 3n
\][/tex]
From this equation, we get:
[tex]\[
\frac{m}{n} = \frac{3}{5}
\][/tex]
Therefore, the ratio [tex]\( m:n \)[/tex] is [tex]\( 3:5 \)[/tex].
5. Use the Ratio to Find the x-coordinate:
[tex]\[
x = \frac{m x_B + n x_A}{m + n}
\][/tex]
Substitute [tex]\( m = 3 \)[/tex], [tex]\( n = 5 \)[/tex], [tex]\( x_A = 6 \)[/tex], and [tex]\( x_B = 4 \)[/tex]:
[tex]\[
x = \frac{3 \cdot 4 + 5 \cdot 6}{3 + 5}
\][/tex]
[tex]\[
x = \frac{12 + 30}{8}
\][/tex]
[tex]\[
x = \frac{42}{8}
\][/tex]
[tex]\[
x = 5.25
\][/tex]
6. Combine the Coordinates to Get the Dividing Point:
The point dividing the line segment joining [tex]\( A(6,5) \)[/tex] and [tex]\( B(4,-3) \)[/tex] at [tex]\( y = 2 \)[/tex] is:
[tex]\[
(5.25, 2)
\][/tex]
So, the coordinates of the point that divides the line segment [tex]\( AB \)[/tex] at [tex]\( y = 2 \)[/tex] are [tex]\( \boxed{(5.25, 2)} \)[/tex].
