To find the coordinates of the point that divides the line segment joining the points [tex]\((-1, 4)\)[/tex] and [tex]\((0, -3)\)[/tex] in the ratio [tex]\(1:4\)[/tex] internally, we will use the section formula.
The section formula states that if a point [tex]\(P(x, y)\)[/tex] divides the line segment joining two points [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 point [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]\(A = (-1, 4)\)[/tex]
- [tex]\(B = (0, -3)\)[/tex]
- Ratio [tex]\(m:n = 1:4\)[/tex]
Using these values in the section formula:
1. Let's find the x-coordinate of the point:
[tex]\[
x = \frac{(1 \cdot 0) + (4 \cdot -1)}{1 + 4} = \frac{0 + (-4)}{5} = \frac{-4}{5} = -0.8
\][/tex]
2. Now, let's find the y-coordinate of the point:
[tex]\[
y = \frac{(1 \cdot -3) + (4 \cdot 4)}{1 + 4} = \frac{(-3) + 16}{5} = \frac{13}{5} = 2.6
\][/tex]
Thus, the coordinates of the point that divides the segment joining [tex]\((-1, 4)\)[/tex] and [tex]\((0, -3)\)[/tex] in the ratio [tex]\(1:4\)[/tex] internally are:
[tex]\[
(-0.8, 2.6)
\][/tex]
