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 use the section formula for internal division. The section formula states that if a point [tex]\(P(x, y)\)[/tex] divides the line segment joining the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in the ratio [tex]\(m:n\)[/tex], then the coordinates of point [tex]\(P\)[/tex] are given by:
[tex]\[
x = \frac{mx_2 + nx_1}{m+n}
\][/tex]
[tex]\[
y = \frac{my_2 + ny_1}{m+n}
\][/tex]
Here, the coordinates of the points are:
[tex]\[
(x_1, y_1) = (-1, 4)
\][/tex]
[tex]\[
(x_2, y_2) = (0, -3)
\][/tex]
The ratio in which the line segment is divided is:
[tex]\[
m:n = 1:4
\][/tex]
So, [tex]\(m = 1\)[/tex] and [tex]\(n = 4\)[/tex].
We will first find the x-coordinate:
[tex]\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m+n}
= \frac{1 \cdot 0 + 4 \cdot (-1)}{1+4}
= \frac{0 - 4}{5}
= \frac{-4}{5}
= -0.8
\][/tex]
Next, we find the y-coordinate:
[tex]\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m+n}
= \frac{1 \cdot (-3) + 4 \cdot 4}{1+4}
= \frac{-3 + 16}{5}
= \frac{13}{5}
= 2.6
\][/tex]
Thus, the coordinates of the point that divides the line 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].
