To determine the ratio in which the point [tex]\((x, 0)\)[/tex] divides the line segment joining the points [tex]\(A(4, 12)\)[/tex] and [tex]\(B(-3, 5)\)[/tex], as well as the corresponding [tex]\(x\)[/tex]-coordinate, we will follow these steps:
1. Find the ratio using the section formula:
The key idea is to use the section formula which states that if a point [tex]\(P(x, y)\)[/tex] divides the line joining [tex]\(A(x_1, y_1)\)[/tex] and [tex]\(B(x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex], then the coordinates of [tex]\(P\)[/tex] can be found using:
[tex]\[
P(x, y) = \left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)
\][/tex]
Since the point [tex]\(P\)[/tex] lies on the x-axis, its coordinates are [tex]\((x, 0)\)[/tex]. Therefore, the y-coordinate equation should satisfy:
[tex]\[
\frac{m y_2 + n y_1}{m + n} = 0
\][/tex]
Plugging in the coordinates of [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
[tex]\[
\frac{m \cdot 5 + n \cdot 12}{m + n} = 0
\][/tex]
Simplifying this equation:
[tex]\[
5m + 12n = 0
\][/tex]
So:
[tex]\[
12n = -5m \quad \Rightarrow \quad \frac{m}{n} = -\frac{12}{5}
\][/tex]
This implies that the ratio [tex]\(m:n = -12:5\)[/tex].
2. Determine the x-coordinate:
Using the same section formula for the x-coordinate:
[tex]\[
x = \frac{m x_2 + n x_1}{m + n}
\][/tex]
Substituting the values [tex]\(x_1 = 4\)[/tex], [tex]\(x_2 = -3\)[/tex], and the ratio [tex]\(m = -12\)[/tex], [tex]\(n = 5\)[/tex]:
[tex]\[
x = \frac{-12 \cdot (-3) + 5 \cdot 4}{-12 + 5}
\][/tex]
Simplifying the numerator:
[tex]\[
-12 \cdot (-3) + 5 \cdot 4 = 36 + 20 = 56
\][/tex]
And the denominator:
[tex]\[
-12 + 5 = -7
\][/tex]
So the value of [tex]\(x\)[/tex] is:
[tex]\[
x = \frac{56}{-7} = -8
\][/tex]
Therefore, the point [tex]\((x, 0)\)[/tex] divides the line segment joining points [tex]\(A(4, 12)\)[/tex] and [tex]\(B(-3, 5)\)[/tex] in the ratio [tex]\(-12:5\)[/tex], and the value of [tex]\(x\)[/tex] is [tex]\(-8\)[/tex].
