To find the coordinates of the point that divides the line segment joining the points [tex]\((-3, -4)\)[/tex] and [tex]\( (2, 1) \)[/tex] in the ratio [tex]\( 3:2 \)[/tex] externally, we can use the formula for external division.
Given points:
- [tex]\( A(x_1, y_1) = (-3, -4) \)[/tex]
- [tex]\( B(x_2, y_2) = (2, 1) \)[/tex]
- The ratio [tex]\( m:n = 3:2 \)[/tex]
The coordinates [tex]\( (x, y) \)[/tex] of the point that divides the line segment externally can be given by the section formula for external division:
[tex]\[
x = \frac{m x_2 - n x_1}{m - n}
\][/tex]
[tex]\[
y = \frac{m y_2 - n y_1}{m - n}
\][/tex]
Substitute the given values into the formulas:
[tex]\[
x = \frac{3 \cdot 2 - 2 \cdot (-3)}{3 - 2}
\][/tex]
[tex]\[
x = \frac{6 + 6}{1}
\][/tex]
[tex]\[
x = \frac{12}{1}
\][/tex]
[tex]\[
x = 12
\][/tex]
Similarly, for the [tex]\( y \)[/tex]-coordinate:
[tex]\[
y = \frac{3 \cdot 1 - 2 \cdot (-4)}{3 - 2}
\][/tex]
[tex]\[
y = \frac{3 + 8}{1}
\][/tex]
[tex]\[
y = \frac{11}{1}
\][/tex]
[tex]\[
y = 11
\][/tex]
Therefore, the coordinates of the point which divides the line segment joining the points [tex]\((-3, -4)\)[/tex] and [tex]\( (2, 1) \)[/tex] in the ratio [tex]\( 3:2 \)[/tex] externally are [tex]\( \left( 12, 11 \right) \)[/tex].
