To find the ratio in which the point [tex]\( P(3, y) \)[/tex] divides the line segment joining the points [tex]\( A(9, 8) \)[/tex] and [tex]\( B(-4, -6) \)[/tex], we can use the section formula.
Section Formula:
If a point [tex]\( P(x, y) \)[/tex] divides the line segment joining points [tex]\( A(x_1, y_1) \)[/tex] and [tex]\( B(x_2, y_2) \)[/tex] in the ratio [tex]\( m:n \)[/tex],
the coordinates [tex]\( P \)[/tex] are given by:
[tex]\[
P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right)
\][/tex]
Given [tex]\( A(9, 8) \)[/tex], [tex]\( B(-4, -6) \)[/tex], and [tex]\( P(3, y) \)[/tex], we equate the x-coordinates first to find the ratio [tex]\( m : n \)[/tex].
Let's denote the ratio as [tex]\( k = \frac{m}{n} \)[/tex].
Equating the x-coordinates:
[tex]\[
\frac{m(-4) + n(9)}{m+n} = 3
\][/tex]
Multiplying through by [tex]\( m + n \)[/tex] to clear the denominator:
[tex]\[
-4m + 9n = 3(m + n)
\][/tex]
Expanding and simplifying:
[tex]\[
-4m + 9n = 3m + 3n
\][/tex]
[tex]\[
-4m - 3m + 9n - 3n = 0
\][/tex]
[tex]\[
-7m + 6n = 0
\][/tex]
Solving for [tex]\( \frac{m}{n} \)[/tex]:
[tex]\[
7m = 6n
\][/tex]
[tex]\[
\frac{m}{n} = \frac{6}{7}
\][/tex]
Thus, the ratio [tex]\( m : n \)[/tex] is [tex]\( 6 : 7 \)[/tex].
Now, to find [tex]\( y \)[/tex], we apply the same ratio to the y-coordinates using the section formula:
[tex]\[
y = \frac{m(-6) + n(8)}{m + n}
\][/tex]
Substituting [tex]\( m = 6 \)[/tex] and [tex]\( n = 7 \)[/tex]:
[tex]\[
y = \frac{6(-6) + 7(8)}{6 + 7}
\][/tex]
Simplifying the numerator and denominator:
[tex]\[
y = \frac{6(-6) + 7(8)}{13}
\][/tex]
[tex]\[
y = \frac{-36 + 56}{13}
\][/tex]
[tex]\[
y = \frac{20}{13}
\][/tex]
Thus, [tex]\( y = \frac{20}{13} \)[/tex].
Conclusion:
The point [tex]\( (3, y) \)[/tex] divides the line segment joining [tex]\( (9, 8) \)[/tex] and [tex]\( (-4, -6) \)[/tex] in the ratio [tex]\( 6 : 7 \)[/tex]. The corresponding value of [tex]\( y \)[/tex] is [tex]\( \frac{20}{13} \)[/tex].
