Question

Determine the location of point L if it dives line A(0, 0)B(-2, 3) into two segments in the ratio 1:3.

Answer 1

To find the coordinates of point \( L \) that divides the line segment \( AB \) in the ratio \( 1:3 \), we can use the section formula.

Let the coordinates of points \( A \) and \( B \) be:
- \( A(0, 0) \)
- \( B(-2, 3) \)

If point \( L \) divides the segment \( AB \) in the ratio \( m:n \) where \( m = 1 \) and \( n = 3 \), we can use the section formula which states:

\[
L\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Here,
- \( (x_1, y_1) = (0, 0) \) (coordinates of \( A \))
- \( (x_2, y_2) = (-2, 3) \) (coordinates of \( B \))
- \( m = 1 \)
- \( n = 3 \)

Applying the formula:

\[
L_x = \frac{1 \cdot (-2) + 3 \cdot (0)}{1 + 3} = \frac{-2 + 0}{4} = \frac{-2}{4} = -\frac{1}{2}
\]

\[
L_y = \frac{1 \cdot 3 + 3 \cdot 0}{1 + 3} = \frac{3 + 0}{4} = \frac{3}{4}
\]

Thus, the coordinates of point \( L \) are:

\[
L\left(-\frac{1}{2}, \frac{3}{4}\right)
\]

Therefore, the location of point \( L \) is:

\[
\boxed{\left(-\frac{1}{2}, \frac{3}{4}\right)}
\]