Sure, let's solve this step-by-step.
We have a line segment joining two points [tex]\((a, b)\)[/tex] and [tex]\((2, 7)\)[/tex] that is divided by a point [tex]\((-2, -9)\)[/tex] in the ratio of 3:4.
To find the coordinates [tex]\((a, b)\)[/tex], we'll use the section formula for internal division of a line segment. The section formula states that if a point [tex]\((x, y)\)[/tex] divides the line segment joining [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex], then the coordinates [tex]\((x, y)\)[/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:
- The coordinates [tex]\((x_2, y_2) = (2, 7)\)[/tex]
- The coordinates [tex]\((x, y) = (-2, -9)\)[/tex]
- The ratio [tex]\(m:n = 3:4\)[/tex]
Let’s denote:
- [tex]\((x_1, y_1) = (a, b)\)[/tex]
- The point [tex]\((x, y) = (-2, -9)\)[/tex]
We will set up equations using the section formula and the given point [tex]\((-2, -9)\)[/tex]:
1. For the x-coordinate:
[tex]\[
-2 = \frac{3 \cdot 2 + 4 \cdot a}{3 + 4}
\][/tex]
2. For the y-coordinate:
[tex]\[
-9 = \frac{3 \cdot 7 + 4 \cdot b}{3 + 4}
\][/tex]
Now, let's solve these equations for [tex]\(a\)[/tex] and [tex]\(b\)[/tex].
Solving for [tex]\(a\)[/tex]:
[tex]\[ -2 = \frac{6 + 4a}{7} \][/tex]
Multiply both sides by 7:
[tex]\[ -14 = 6 + 4a \][/tex]
Subtract 6 from both sides:
[tex]\[ -20 = 4a \][/tex]
Divide by 4:
[tex]\[ a = -5 \][/tex]
Solving for [tex]\(b\)[/tex]:
[tex]\[ -9 = \frac{21 + 4b}{7} \][/tex]
Multiply both sides by 7:
[tex]\[ -63 = 21 + 4b \][/tex]
Subtract 21 from both sides:
[tex]\[ -84 = 4b \][/tex]
Divide by 4:
[tex]\[ b = -21 \][/tex]
So, the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are:
[tex]\[ a = -5 \][/tex]
[tex]\[ b = -21 \][/tex]
Thus, the coordinates [tex]\((a, b)\)[/tex] are [tex]\((-5, -21)\)[/tex].
