Sure, let's solve the given problem step-by-step.
### Given:
- Point [tex]\( M(6, 5) \)[/tex]
- Point [tex]\( N(-3, 2) \)[/tex]
- Point [tex]\( A(x, 4) \)[/tex] divides the segment [tex]\( MN \)[/tex].
### (i) Finding the ratio [tex]\( m:n \)[/tex] in which [tex]\( A \)[/tex] divides [tex]\( MN \)[/tex]:
By the section formula, if a point [tex]\( A(x, y) \)[/tex] divides the line segment joining points [tex]\( M(x_1, y_1) \)[/tex] and [tex]\( N(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex], then the coordinates of [tex]\( A \)[/tex] are given by:
[tex]\[ x = \frac{mx_2 + nx_1}{m + n} \][/tex]
[tex]\[ y = \frac{my_2 + ny_1}{m + n} \][/tex]
Given that [tex]\( y = 4 \)[/tex] at point [tex]\( A(x, 4) \)[/tex],
[tex]\[ 4 = \frac{m \cdot 2 + n \cdot 5}{m + n} \][/tex]
Solving for the ratio:
[tex]\[ 4(m + n) = 2m + 5n \][/tex]
[tex]\[ 4m + 4n = 2m + 5n \][/tex]
[tex]\[ 4m - 2m = 5n - 4n \][/tex]
[tex]\[ 2m = n \][/tex]
[tex]\[ n = 2m \][/tex]
Therefore, the ratio [tex]\( m:n = 1:2 \)[/tex].
### (ii) Finding the value of [tex]\( x \)[/tex]:
Using the ratio [tex]\( 1:2 \)[/tex] in the section formula for the [tex]\( x \)[/tex]-coordinate:
[tex]\[ x = \frac{mx_2 + nx_1}{m + n} \][/tex]
Here, [tex]\( x_1 = 6 \)[/tex] and [tex]\( x_2 = -3 \)[/tex], and [tex]\( m=1 \)[/tex] and [tex]\( n=2 \)[/tex],
[tex]\[ x = \frac{1 \cdot (-3) + 2 \cdot 6}{1 + 2} \][/tex]
[tex]\[ x = \frac{-3 + 12}{3} \][/tex]
[tex]\[ x = \frac{9}{3} \][/tex]
[tex]\[ x = 3 \][/tex]
### Final Answer:
(i) The ratio in which point [tex]\( A \)[/tex] divides the line segment [tex]\( MN \)[/tex] is [tex]\( 1:2 \)[/tex].
(ii) The value of [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex].
